{"id":9223,"date":"2017-05-02T19:41:37","date_gmt":"2017-05-02T19:41:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9223"},"modified":"2024-04-29T18:41:55","modified_gmt":"2024-04-29T18:41:55","slug":"multiplying-whole-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/multiplying-whole-numbers\/","title":{"raw":"Multiplying Whole Numbers","rendered":"Multiplying Whole Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify and use the multiplication property of zero<\/li>\r\n \t<li>Identify and use the identity property of multiplication<\/li>\r\n \t<li>Identify and use the commutative property of multiplication<\/li>\r\n \t<li>Multiply multiple-digit numbers using columns for place value<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Multiply Whole Numbers<\/h2>\r\nIn order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.\r\n\r\nEach box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.\r\n<table id=\"fs-id1563789\" class=\"column-header\" style=\"width: 398px;\" summary=\"This is a multiplication table with 11 columns and 11 rows. The first column has the values \">\r\n<thead>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<th style=\"width: 11px; height: 15px;\">[latex]\u00d7[\/latex]<\/th>\r\n<th style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/th>\r\n<th style=\"width: 9px; height: 15px;\">[latex]1[\/latex]<\/th>\r\n<th style=\"width: 16px; height: 15px;\">[latex]2[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]3[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]4[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]5[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]7[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]27[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]20[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]28[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]32[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]20[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]25[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]30[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]35[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]40[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]45[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]30[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]42[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]48[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]54[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.7812px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15.7812px;\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15.7812px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15.7812px;\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15.7812px;\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]28[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]35[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]42[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]49[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]56[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\">[latex]63[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]32[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]40[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]48[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]56[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]64[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]72[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]27[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]45[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]54[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]63[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]72[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\">[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhat happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication Property of Zero<\/h3>\r\nThe product of any number and [latex]0[\/latex] is [latex]0[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ a\\cdot 0=0\\hfill \\\\ 0\\cdot a=0\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply:\r\n<ol>\r\n \t<li>[latex]0\\cdot 11[\/latex]<\/li>\r\n \t<li>[latex]\\left(42\\right)0[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168287095915\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]0\\cdot 11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of any number and zero is zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]\\left(42\\right)0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplying by zero results in zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144419&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhat happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and [latex]1[\/latex] is called the multiplicative identity.\r\n<div class=\"textbox shaded\">\r\n<h3>Identity Property of Multiplication<\/h3>\r\nThe product of any number and [latex]1[\/latex] is the number.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1\\cdot a=a\\\\ a\\cdot 1=a\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply:\r\n<ol>\r\n \t<li>[latex]\\left(11\\right)1[\/latex]<\/li>\r\n \t<li>[latex]1\\cdot 42[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"229212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229212\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168288255207\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]\\left(11\\right)1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product of any number and one is the number.<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]1\\cdot 42[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiplying by one does not change the value.<\/td>\r\n<td>[latex]42[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144421&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nEarlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that [latex]8+9=17[\/latex] is the same as [latex]9+8=17[\/latex].\r\n\r\nIs this also true for multiplication? Let\u2019s look at a few pairs of factors.\r\n<p style=\"text-align: center;\">[latex]4\\cdot 7=28\\quad 7\\cdot 4=28[\/latex]\r\n[latex]9\\cdot 7=63\\quad 7\\cdot 9=63[\/latex]\r\n[latex]8\\cdot 9=72\\quad 9\\cdot 8=72[\/latex]<\/p>\r\nWhen the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Commutative Property of Multiplication<\/h3>\r\nChanging the order of the factors does not change their product.\r\n<p style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply:\r\n\r\n[latex]8\\cdot 7[\/latex]\r\n[latex]7\\cdot 8[\/latex]\r\n\r\n[reveal-answer q=\"468629\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"468629\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168287497846\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]8\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]7\\cdot 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nChanging the order of the factors does not change the product.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]188577[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nTo multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 27\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\end{array}[\/latex]<\/p>\r\nWe start by multiplying [latex]3[\/latex] by [latex]7[\/latex].\r\n<p style=\"text-align: center;\">[latex]3\\times 7=21[\/latex]<\/p>\r\nWe write the [latex]1[\/latex] in the ones place of the product. We carry the [latex]2[\/latex] tens by writing [latex]2[\/latex] above the tens place.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215526\/CNX_BMath_Figure_01_04_008_img.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"165\" height=\"153\" \/>\r\nThen we multiply the [latex]3[\/latex] by the [latex]2[\/latex], and add the [latex]2[\/latex] above the tens place to the product. So [latex]3\\times 2=6[\/latex], and [latex]6+2=8[\/latex]. Write the [latex]8[\/latex] in the tens place of the product.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215527\/CNX_BMath_Figure_01_04_010_img.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"221\" height=\"124\" \/>\r\nThe product is [latex]81[\/latex].<\/p>\r\n&nbsp;\r\n\r\nWhen we multiply two numbers with a different number of digits, it\u2019s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]15\\cdot 4[\/latex]\r\n\r\n[reveal-answer q=\"942773\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"942773\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168287565780\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the digits [latex]5[\/latex] and [latex]4[\/latex] line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 15\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]4[\/latex] by the digit in the ones place of [latex]15[\/latex].\u00a0\u00a0 [latex]4\\cdot 5=20[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]0[\/latex] in the ones place of the product and carry the [latex]2[\/latex] tens.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]4[\/latex] by the digit in the tens place of [latex]15[\/latex].\u00a0 [latex]4\\cdot 1=4[\/latex]. Add the [latex]2[\/latex] tens we carried. [latex]4+2=6[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the [latex]6[\/latex] in the tens place of the product.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 60\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144427&amp;theme=oea&amp;iframe_resize_id=mom[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]286\\cdot 5[\/latex]\r\n\r\n[reveal-answer q=\"849168\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849168\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168288632671\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\r\n<tbody>\r\n<tr style=\"height: 109.441px;\">\r\n<td style=\"height: 109.441px;\">Write the numbers so the digits [latex]5[\/latex] and [latex]6[\/latex] line up vertically.<\/td>\r\n<td style=\"height: 109.441px;\">[latex]\\begin{array}{c}\\hfill 286\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 88px;\">\r\n<td style=\"height: 88px;\">Multiply [latex]5[\/latex] by the digit in the ones place of [latex]286[\/latex].\u00a0 [latex]5\\cdot 6=30[\/latex]<\/td>\r\n<td style=\"height: 88px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 145px;\">\r\n<td style=\"height: 145px;\">Write the [latex]0[\/latex] in the ones place of the product and carry the [latex]3[\/latex] to the tens place. Multiply [latex]5[\/latex] by the digit in the tens place of [latex]286[\/latex].\u00a0 [latex]5\\cdot 8=40[\/latex]<\/td>\r\n<td style=\"height: 145px;\">[latex]\\begin{array}{}\\\\ \\hfill 2\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 161px;\">\r\n<td style=\"height: 161px;\">Add the [latex]3[\/latex] tens we carried to get [latex]40+3=43[\/latex]. Write the [latex]3[\/latex] in the tens place of the product and carry the [latex]4[\/latex] to the hundreds place.<\/td>\r\n<td style=\"height: 161px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 30\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 251px;\">\r\n<td style=\"height: 251px;\">Multiply [latex]5[\/latex] by the digit in the hundreds place of [latex]286[\/latex].\u00a0\u00a0 [latex]5\\cdot 2=10[\/latex]. Add the [latex]4[\/latex] hundreds we carried to get [latex]10+4=14[\/latex].\r\n\r\nWrite the [latex]4[\/latex] in the hundreds place of the product and the [latex]1[\/latex] to the thousands place.<\/td>\r\n<td style=\"height: 251px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 1,430\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144429&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiply two whole numbers to find the product<\/h3>\r\n<ol id=\"eip-id1168286064302\" class=\"stepwise\">\r\n \t<li>Write the numbers so each place value lines up vertically.<\/li>\r\n \t<li>Multiply the digits in each place value.\r\n<ul id=\"eip-id1168288663980\">\r\n \t<li>Work from right to left, starting with the ones place in the bottom number.\r\n<ul id=\"eip-id1168288663983\">\r\n \t<li>Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.<\/li>\r\n \t<li>If a product in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\r\n \t<li>Write the partial products, lining up the digits in the place values with the numbers above.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Repeat for the tens place in the bottom number, the hundreds place, and so on.<\/li>\r\n \t<li>Insert a zero as a placeholder with each additional partial product.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Add the partial products.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]62\\left(87\\right)[\/latex]\r\n\r\n[reveal-answer q=\"698602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698602\"]\r\n\r\nSolution\r\n<table id=\"eip-279\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"No alt text\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Write the numbers so each place lines up vertically.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215528\/CNX_BMath_Figure_01_04_020_img-02.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"46\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62.\u00a0 [latex]7\\cdot 2=14[\/latex].\r\n\r\nWrite the 4 in the ones place of the product and carry the 1 to the tens place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215529\/CNX_BMath_Figure_01_04_020_img-03.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"92\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Multiply 7 by the digit in the tens place of 62: [latex]7\\cdot 6=42[\/latex].\u00a0 Add the 1 ten we carried: [latex]42+1=43[\/latex].\r\n\r\nWrite the 3 in the tens place of the product and the 4 in the hundreds place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-04.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"92\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">The first partial product is [latex]434[\/latex].<\/td>\r\n<td style=\"width: 308.097px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Now, write a [latex]0[\/latex] under the [latex]4[\/latex] in the ones place of the next partial product as a placeholder, since we now multiply the digit in the tens place of [latex]87[\/latex] by [latex]62[\/latex]. Multiply [latex]8[\/latex] by the digit in the ones place of [latex]62[\/latex].\r\n\r\n[latex]8\\cdot 2=16[\/latex]. Write the [latex]6[\/latex] in the next place of the product, which is the tens place. Carry the [latex]1[\/latex] to the tens place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-05.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Multiply [latex]8[\/latex] by [latex]6[\/latex], the digit in the tens place of [latex]62[\/latex], then add the [latex]1[\/latex] ten we carried to get [latex]49[\/latex]. Write the [latex]9[\/latex] in the hundreds place of the product and the [latex]4[\/latex] in the thousands place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215531\/CNX_BMath_Figure_01_04_020_img-06.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">The second partial product is [latex]4960[\/latex]. Add the partial products.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215532\/CNX_BMath_Figure_01_04_020_img-07.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text. \" width=\"42\" height=\"155\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe product is [latex]5,394[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144432&amp;theme=oea&amp;iframe_resize_id=mom6[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply:\r\n<ol>\r\n \t<li>[latex]47\\cdot 10[\/latex]<\/li>\r\n \t<li>[latex]47\\cdot 100[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"880886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880886\"]\r\n\r\nSolution\r\n<table id=\"eip-id1441016\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.[latex]47\\cdot 10[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 47\\\\ \\hfill \\underset{\\text{___}}{\\times 10}\\\\ \\hfill 00\\\\ \\hfill \\underset{\\text{___}}{470}\\\\ \\hfill 470\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.[latex]47\\cdot 100[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 47\\\\ \\hfill \\underset{\\text{_____}}{\\times 100}\\\\ \\hfill 00\\\\ \\hfill \\underset{\\text{_____}}{\\begin{array}{c}\\hfill 000\\\\ \\hfill 4700\\end{array}}\\\\ \\hfill 4,700\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen we multiplied [latex]47[\/latex] times [latex]10[\/latex], the product was [latex]470[\/latex]. Notice that [latex]10[\/latex] has one zero, and we put one zero after [latex]47[\/latex], to get the product. When we multiplied [latex]47[\/latex] times [latex]100[\/latex], the product was [latex]4,700[\/latex]. Notice that [latex]100[\/latex] has two zeros and we put two zeros after [latex]47[\/latex], to get the product.\r\n\r\nDo you see the pattern? If we multiplied [latex]47[\/latex] times [latex]10,000[\/latex], which has four zeros, we would put four zeros after [latex]47[\/latex] to get the product [latex]470,000[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nMultiply:\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144433&amp;theme=oea&amp;iframe_resize_id=mom7[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(354\\right)\\left(438\\right)[\/latex]\r\n\r\n[reveal-answer q=\"301500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"301500\"]\r\n\r\nSolution\r\nThere are three digits in the factors, so there will be [latex]3[\/latex] partial products. We do not have to write the [latex]0[\/latex] as a placeholder, as long as we write each partial product in the correct place.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215533\/CNX_BMath_Figure_01_04_011_img.png\" alt=\"An image of the multiplication problem \u201c354 times 438\u201d worked out vertically. 354 is the top number, 438 is the second number. Below 438 is a multiplication bar. Below the bar is the number 2,832. 2832 has the label \u201cMultiply 8 times 354\u201d. Below 2832 is the number 1,062; 1062 has the label \u201cMultiply 3 times 354\u201d. Below 1062 is the number 1,416; 1416 has the label \u201cMultiply 4 times 354\u201d. Below this is a bar and below the bar is the number \u201c155,052\u201d, with the label \u201cAdd the partial products\u201d.\" width=\"450\" height=\"140\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144434&amp;amp;theme=oea&amp;amp;iframe_resize_id=mom8[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(896\\right)201[\/latex]\r\n\r\n[reveal-answer q=\"58598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"58598\"]\r\n\r\nSolution\r\nThere should be [latex]3[\/latex] partial products. The second partial product will be the result of multiplying [latex]896[\/latex] by [latex]0[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215535\/CNX_BMath_Figure_01_04_012_img.png\" alt=\"An image of the multiplication problem \u201c896 times 201\u201d worked out vertically. 896 is the top number, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 201 is the second number, the 2 in the hundreds place, the 0 in the tens place, the 1 in the ones place. Below 201 is a multiplcation bar. Below the bar is the number 896, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 896 has the label \u201cMultiply 1 times 896\u201d. Below 896 is the number \u201c000\u201d, the 0 in the thousands place, the 0 in the hundreds place, and the 0 in the tens place. \u201c000\u201d has the label \u201cMultiply 0 times 896\u201d. Below \u201c000\u201d is the number 1792, the 1 in the hundred thousands place, the 7 in the ten thousands place, the 9 in the thousands place, and the 2 in the hundreds place. 1792 has the label \u201cMultiply 2 times 896\u201d. Below this is a bar and below the bar is the number \u201c180,096\u201d, with the label \u201cAdd the partial products\u201d.\" width=\"450\" height=\"140\" \/>\r\nNotice that the second partial product of all zeros doesn\u2019t really affect the result. We can place a zero as a placeholder in the tens place, and then proceed directly to multiplying by the [latex]2[\/latex] in the hundreds place, as shown.\r\nMultiply by [latex]10[\/latex], but insert only one zero as a placeholder in the tens place. Multiply by [latex]200[\/latex], putting the [latex]2[\/latex] from the [latex]12[\/latex] ([latex]2\\cdot 6=12[\/latex]) in the hundreds place.\r\n\r\n[latex]\\begin{array}{}\\\\ \\\\ \\hfill 896\\\\ \\hfill \\underset{\\text{_____}}{\\times 201}\\\\ \\hfill 896\\\\ \\hfill \\underset{\\text{__________}}{17920}\\\\ \\hfill 180,096\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144434&amp;theme=oea&amp;iframe_resize_id=mom9[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen there are three or more factors, we multiply the first two, and then multiply their product by the next factor. For example:\r\n<table id=\"eip-0\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>to multiply<\/td>\r\n<td>[latex]8\\cdot 3\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>first multiply [latex]8\\cdot 3[\/latex]<\/td>\r\n<td>[latex]24\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>then multiply [latex]24\\cdot 2[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the video below, we summarize the concepts presented on this page, including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.\r\n\r\nhttps:\/\/youtu.be\/kW7JBfplJGE","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify and use the multiplication property of zero<\/li>\n<li>Identify and use the identity property of multiplication<\/li>\n<li>Identify and use the commutative property of multiplication<\/li>\n<li>Multiply multiple-digit numbers using columns for place value<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Multiply Whole Numbers<\/h2>\n<p>In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.<\/p>\n<p>Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.<\/p>\n<table id=\"fs-id1563789\" class=\"column-header\" style=\"width: 398px;\" summary=\"This is a multiplication table with 11 columns and 11 rows. The first column has the values\">\n<thead>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<th style=\"width: 11px; height: 15px;\">[latex]\u00d7[\/latex]<\/th>\n<th style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/th>\n<th style=\"width: 9px; height: 15px;\">[latex]1[\/latex]<\/th>\n<th style=\"width: 16px; height: 15px;\">[latex]2[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]3[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]4[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]5[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]7[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]7[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]14[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]15[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]21[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]27[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]20[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]28[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]32[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]10[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]15[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]20[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]25[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]30[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]35[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]40[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]45[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]6[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]30[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]42[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]48[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]54[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.7812px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15.7812px;\">[latex]7[\/latex]<\/td>\n<td style=\"width: 122px; height: 15.7812px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15.7812px;\">[latex]7[\/latex]<\/td>\n<td style=\"width: 16px; height: 15.7812px;\">[latex]14[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]21[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]28[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]35[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]42[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]49[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]56[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\">[latex]63[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]32[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]40[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]48[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]56[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]64[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]72[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\">[latex]9[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]27[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]36[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]45[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]54[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]63[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]72[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\">[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplication Property of Zero<\/h3>\n<p>The product of any number and [latex]0[\/latex] is [latex]0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ a\\cdot 0=0\\hfill \\\\ 0\\cdot a=0\\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply:<\/p>\n<ol>\n<li>[latex]0\\cdot 11[\/latex]<\/li>\n<li>[latex]\\left(42\\right)0[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168287095915\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]0\\cdot 11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of any number and zero is zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]\\left(42\\right)0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiplying by zero results in zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144419\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144419&#38;theme=oea&#38;iframe_resize_id=ohm144419&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and [latex]1[\/latex] is called the multiplicative identity.<\/p>\n<div class=\"textbox shaded\">\n<h3>Identity Property of Multiplication<\/h3>\n<p>The product of any number and [latex]1[\/latex] is the number.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}1\\cdot a=a\\\\ a\\cdot 1=a\\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply:<\/p>\n<ol>\n<li>[latex]\\left(11\\right)1[\/latex]<\/li>\n<li>[latex]1\\cdot 42[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q229212\">Show Solution<\/span><\/p>\n<div id=\"q229212\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168288255207\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]\\left(11\\right)1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product of any number and one is the number.<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]1\\cdot 42[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiplying by one does not change the value.<\/td>\n<td>[latex]42[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144421\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144421&#38;theme=oea&#38;iframe_resize_id=ohm144421&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that [latex]8+9=17[\/latex] is the same as [latex]9+8=17[\/latex].<\/p>\n<p>Is this also true for multiplication? Let\u2019s look at a few pairs of factors.<\/p>\n<p style=\"text-align: center;\">[latex]4\\cdot 7=28\\quad 7\\cdot 4=28[\/latex]<br \/>\n[latex]9\\cdot 7=63\\quad 7\\cdot 9=63[\/latex]<br \/>\n[latex]8\\cdot 9=72\\quad 9\\cdot 8=72[\/latex]<\/p>\n<p>When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Commutative Property of Multiplication<\/h3>\n<p>Changing the order of the factors does not change their product.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply:<\/p>\n<p>[latex]8\\cdot 7[\/latex]<br \/>\n[latex]7\\cdot 8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q468629\">Show Solution<\/span><\/p>\n<div id=\"q468629\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168287497846\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]8\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]7\\cdot 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]56[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Changing the order of the factors does not change the product.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm188577\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=188577&theme=oea&iframe_resize_id=ohm188577&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 27\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\end{array}[\/latex]<\/p>\n<p>We start by multiplying [latex]3[\/latex] by [latex]7[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\times 7=21[\/latex]<\/p>\n<p>We write the [latex]1[\/latex] in the ones place of the product. We carry the [latex]2[\/latex] tens by writing [latex]2[\/latex] above the tens place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215526\/CNX_BMath_Figure_01_04_008_img.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"165\" height=\"153\" \/><br \/>\nThen we multiply the [latex]3[\/latex] by the [latex]2[\/latex], and add the [latex]2[\/latex] above the tens place to the product. So [latex]3\\times 2=6[\/latex], and [latex]6+2=8[\/latex]. Write the [latex]8[\/latex] in the tens place of the product.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215527\/CNX_BMath_Figure_01_04_010_img.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"221\" height=\"124\" \/><br \/>\nThe product is [latex]81[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>When we multiply two numbers with a different number of digits, it\u2019s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]15\\cdot 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q942773\">Show Solution<\/span><\/p>\n<div id=\"q942773\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168287565780\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>Write the numbers so the digits [latex]5[\/latex] and [latex]4[\/latex] line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 15\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]4[\/latex] by the digit in the ones place of [latex]15[\/latex].\u00a0\u00a0 [latex]4\\cdot 5=20[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]0[\/latex] in the ones place of the product and carry the [latex]2[\/latex] tens.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]4[\/latex] by the digit in the tens place of [latex]15[\/latex].\u00a0 [latex]4\\cdot 1=4[\/latex]. Add the [latex]2[\/latex] tens we carried. [latex]4+2=6[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the [latex]6[\/latex] in the tens place of the product.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 60\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144427\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144427&#38;theme=oea&#38;iframe_resize_id=ohm144427&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]286\\cdot 5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849168\">Show Solution<\/span><\/p>\n<div id=\"q849168\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288632671\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\n<tbody>\n<tr style=\"height: 109.441px;\">\n<td style=\"height: 109.441px;\">Write the numbers so the digits [latex]5[\/latex] and [latex]6[\/latex] line up vertically.<\/td>\n<td style=\"height: 109.441px;\">[latex]\\begin{array}{c}\\hfill 286\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 88px;\">\n<td style=\"height: 88px;\">Multiply [latex]5[\/latex] by the digit in the ones place of [latex]286[\/latex].\u00a0 [latex]5\\cdot 6=30[\/latex]<\/td>\n<td style=\"height: 88px;\"><\/td>\n<\/tr>\n<tr style=\"height: 145px;\">\n<td style=\"height: 145px;\">Write the [latex]0[\/latex] in the ones place of the product and carry the [latex]3[\/latex] to the tens place. Multiply [latex]5[\/latex] by the digit in the tens place of [latex]286[\/latex].\u00a0 [latex]5\\cdot 8=40[\/latex]<\/td>\n<td style=\"height: 145px;\">[latex]\\begin{array}{}\\\\ \\hfill 2\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 161px;\">\n<td style=\"height: 161px;\">Add the [latex]3[\/latex] tens we carried to get [latex]40+3=43[\/latex]. Write the [latex]3[\/latex] in the tens place of the product and carry the [latex]4[\/latex] to the hundreds place.<\/td>\n<td style=\"height: 161px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 30\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 251px;\">\n<td style=\"height: 251px;\">Multiply [latex]5[\/latex] by the digit in the hundreds place of [latex]286[\/latex].\u00a0\u00a0 [latex]5\\cdot 2=10[\/latex]. Add the [latex]4[\/latex] hundreds we carried to get [latex]10+4=14[\/latex].<\/p>\n<p>Write the [latex]4[\/latex] in the hundreds place of the product and the [latex]1[\/latex] to the thousands place.<\/td>\n<td style=\"height: 251px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 1,430\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144429\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144429&#38;theme=oea&#38;iframe_resize_id=ohm144429&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiply two whole numbers to find the product<\/h3>\n<ol id=\"eip-id1168286064302\" class=\"stepwise\">\n<li>Write the numbers so each place value lines up vertically.<\/li>\n<li>Multiply the digits in each place value.\n<ul id=\"eip-id1168288663980\">\n<li>Work from right to left, starting with the ones place in the bottom number.\n<ul id=\"eip-id1168288663983\">\n<li>Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.<\/li>\n<li>If a product in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\n<li>Write the partial products, lining up the digits in the place values with the numbers above.<\/li>\n<\/ul>\n<\/li>\n<li>Repeat for the tens place in the bottom number, the hundreds place, and so on.<\/li>\n<li>Insert a zero as a placeholder with each additional partial product.<\/li>\n<\/ul>\n<\/li>\n<li>Add the partial products.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]62\\left(87\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698602\">Show Solution<\/span><\/p>\n<div id=\"q698602\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-279\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"No alt text\">\n<tbody>\n<tr>\n<td style=\"width: 585.903px;\">Write the numbers so each place lines up vertically.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215528\/CNX_BMath_Figure_01_04_020_img-02.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"46\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62.\u00a0 [latex]7\\cdot 2=14[\/latex].<\/p>\n<p>Write the 4 in the ones place of the product and carry the 1 to the tens place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215529\/CNX_BMath_Figure_01_04_020_img-03.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"92\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Multiply 7 by the digit in the tens place of 62: [latex]7\\cdot 6=42[\/latex].\u00a0 Add the 1 ten we carried: [latex]42+1=43[\/latex].<\/p>\n<p>Write the 3 in the tens place of the product and the 4 in the hundreds place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-04.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"92\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">The first partial product is [latex]434[\/latex].<\/td>\n<td style=\"width: 308.097px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Now, write a [latex]0[\/latex] under the [latex]4[\/latex] in the ones place of the next partial product as a placeholder, since we now multiply the digit in the tens place of [latex]87[\/latex] by [latex]62[\/latex]. Multiply [latex]8[\/latex] by the digit in the ones place of [latex]62[\/latex].<\/p>\n<p>[latex]8\\cdot 2=16[\/latex]. Write the [latex]6[\/latex] in the next place of the product, which is the tens place. Carry the [latex]1[\/latex] to the tens place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-05.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Multiply [latex]8[\/latex] by [latex]6[\/latex], the digit in the tens place of [latex]62[\/latex], then add the [latex]1[\/latex] ten we carried to get [latex]49[\/latex]. Write the [latex]9[\/latex] in the hundreds place of the product and the [latex]4[\/latex] in the thousands place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215531\/CNX_BMath_Figure_01_04_020_img-06.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"127\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">The second partial product is [latex]4960[\/latex]. Add the partial products.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215532\/CNX_BMath_Figure_01_04_020_img-07.png\" alt=\"Appropriate alternative text for this image can be found in the preceding text.\" width=\"42\" height=\"155\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The product is [latex]5,394[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144432\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144432&#38;theme=oea&#38;iframe_resize_id=ohm144432&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply:<\/p>\n<ol>\n<li>[latex]47\\cdot 10[\/latex]<\/li>\n<li>[latex]47\\cdot 100[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880886\">Show Solution<\/span><\/p>\n<div id=\"q880886\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1441016\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\".\">\n<tbody>\n<tr>\n<td>1.[latex]47\\cdot 10[\/latex]<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 47\\\\ \\hfill \\underset{\\text{___}}{\\times 10}\\\\ \\hfill 00\\\\ \\hfill \\underset{\\text{___}}{470}\\\\ \\hfill 470\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.[latex]47\\cdot 100[\/latex]<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 47\\\\ \\hfill \\underset{\\text{_____}}{\\times 100}\\\\ \\hfill 00\\\\ \\hfill \\underset{\\text{_____}}{\\begin{array}{c}\\hfill 000\\\\ \\hfill 4700\\end{array}}\\\\ \\hfill 4,700\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When we multiplied [latex]47[\/latex] times [latex]10[\/latex], the product was [latex]470[\/latex]. Notice that [latex]10[\/latex] has one zero, and we put one zero after [latex]47[\/latex], to get the product. When we multiplied [latex]47[\/latex] times [latex]100[\/latex], the product was [latex]4,700[\/latex]. Notice that [latex]100[\/latex] has two zeros and we put two zeros after [latex]47[\/latex], to get the product.<\/p>\n<p>Do you see the pattern? If we multiplied [latex]47[\/latex] times [latex]10,000[\/latex], which has four zeros, we would put four zeros after [latex]47[\/latex] to get the product [latex]470,000[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Multiply:<\/p>\n<p><iframe loading=\"lazy\" id=\"ohm144433\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144433&#38;theme=oea&#38;iframe_resize_id=ohm144433&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(354\\right)\\left(438\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q301500\">Show Solution<\/span><\/p>\n<div id=\"q301500\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThere are three digits in the factors, so there will be [latex]3[\/latex] partial products. We do not have to write the [latex]0[\/latex] as a placeholder, as long as we write each partial product in the correct place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215533\/CNX_BMath_Figure_01_04_011_img.png\" alt=\"An image of the multiplication problem \u201c354 times 438\u201d worked out vertically. 354 is the top number, 438 is the second number. Below 438 is a multiplication bar. Below the bar is the number 2,832. 2832 has the label \u201cMultiply 8 times 354\u201d. Below 2832 is the number 1,062; 1062 has the label \u201cMultiply 3 times 354\u201d. Below 1062 is the number 1,416; 1416 has the label \u201cMultiply 4 times 354\u201d. Below this is a bar and below the bar is the number \u201c155,052\u201d, with the label \u201cAdd the partial products\u201d.\" width=\"450\" height=\"140\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144434\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144434&#38;theme=oea&#38;iframe_resize_id=ohm144434&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(896\\right)201[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q58598\">Show Solution<\/span><\/p>\n<div id=\"q58598\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThere should be [latex]3[\/latex] partial products. The second partial product will be the result of multiplying [latex]896[\/latex] by [latex]0[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215535\/CNX_BMath_Figure_01_04_012_img.png\" alt=\"An image of the multiplication problem \u201c896 times 201\u201d worked out vertically. 896 is the top number, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 201 is the second number, the 2 in the hundreds place, the 0 in the tens place, the 1 in the ones place. Below 201 is a multiplcation bar. Below the bar is the number 896, the 8 in the hundreds place, the 9 in the tens place, the 6 in the ones place. 896 has the label \u201cMultiply 1 times 896\u201d. Below 896 is the number \u201c000\u201d, the 0 in the thousands place, the 0 in the hundreds place, and the 0 in the tens place. \u201c000\u201d has the label \u201cMultiply 0 times 896\u201d. Below \u201c000\u201d is the number 1792, the 1 in the hundred thousands place, the 7 in the ten thousands place, the 9 in the thousands place, and the 2 in the hundreds place. 1792 has the label \u201cMultiply 2 times 896\u201d. Below this is a bar and below the bar is the number \u201c180,096\u201d, with the label \u201cAdd the partial products\u201d.\" width=\"450\" height=\"140\" \/><br \/>\nNotice that the second partial product of all zeros doesn\u2019t really affect the result. We can place a zero as a placeholder in the tens place, and then proceed directly to multiplying by the [latex]2[\/latex] in the hundreds place, as shown.<br \/>\nMultiply by [latex]10[\/latex], but insert only one zero as a placeholder in the tens place. Multiply by [latex]200[\/latex], putting the [latex]2[\/latex] from the [latex]12[\/latex] ([latex]2\\cdot 6=12[\/latex]) in the hundreds place.<\/p>\n<p>[latex]\\begin{array}{}\\\\ \\\\ \\hfill 896\\\\ \\hfill \\underset{\\text{_____}}{\\times 201}\\\\ \\hfill 896\\\\ \\hfill \\underset{\\text{__________}}{17920}\\\\ \\hfill 180,096\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144434\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144434&#38;theme=oea&#38;iframe_resize_id=ohm144434&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When there are three or more factors, we multiply the first two, and then multiply their product by the next factor. For example:<\/p>\n<table id=\"eip-0\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>to multiply<\/td>\n<td>[latex]8\\cdot 3\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>first multiply [latex]8\\cdot 3[\/latex]<\/td>\n<td>[latex]24\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>then multiply [latex]24\\cdot 2[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the video below, we summarize the concepts presented on this page, including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiplying Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/kW7JBfplJGE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9223\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Multiplying Whole Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/kW7JBfplJGE\">https:\/\/youtu.be\/kW7JBfplJGE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 144419, 144421, 144424, 144427, 144429, 144432, 144433, 144434, 144434. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Multiplying Whole Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/kW7JBfplJGE\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Question ID: 144419, 144421, 144424, 144427, 144429, 144432, 144433, 144434, 144434\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"c5f1427cacd340e58458e7b7d5cbc059, 76a76ab19077433199cf53f9e014bc8b, 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