{"id":3919,"date":"2020-02-11T03:34:13","date_gmt":"2020-02-11T03:34:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3919"},"modified":"2021-02-05T23:56:59","modified_gmt":"2021-02-05T23:56:59","slug":"multiplying-decimals","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/multiplying-decimals\/","title":{"raw":"Multiplying Decimals","rendered":"Multiplying Decimals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply two decimals together<\/li>\r\n \t<li>Multiply a decimal by 10, 100, or 1000<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p>Multiplying decimals is very much like multiplying whole numbers\u2014we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.<\/p>\r\n<p>Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.<\/p>\r\n<p>So let\u2019s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side below. Look for a pattern.<\/p>\r\n\r\n<table id=\"fs-id1620547\" summary=\"A table is shown with 3 columns and 5 rows. The first row is a header row. The first column is not labeled. The other columns are labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>A<\/th>\r\n<th>B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\left(0.3\\right)\\left(0.7\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(0.2\\right)\\left(0.46\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Convert to fractions.<\/td>\r\n<td>[latex]\\left({\\Large\\frac{3}{10}}\\right)\\left({\\Large\\frac{7}{10}}\\right)[\/latex]<\/td>\r\n<td>[latex]\\left({\\Large\\frac{2}{10}}\\right)\\left({\\Large\\frac{46}{100}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Multiply.<\/td>\r\n<td>[latex]{\\Large\\frac{21}{100}}[\/latex]<\/td>\r\n<td>[latex]{\\Large\\frac{92}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Convert back to decimals.<\/td>\r\n<td>[latex]0.21[\/latex]<\/td>\r\n<td>[latex]0.092[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.\r\n\r\nHow many decimal places would you expect for the product of [latex]\\left(0.01\\right)\\left(0.004\\right)?[\/latex] If you said \"five\", you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors\u2014in this case two plus three\u2014to get the number of decimal places in the product\u2014in this case five.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221548\/CNX_BMath_Figure_05_02_018_img.png\" alt=\"The top line says 0.01 times 0.004 equals 0.00004. Below the 0.01, it says 2 places. Below the 0.004, it says 3 places. Below the 0.00004, it says 5 places. The bottom line says 1 over 100 times 4 over 1000 equals 4 over 100,000.\" \/>\r\nOnce we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.\r\n\r\nThe rules for multiplying positive and negative numbers apply to decimals, too, of course.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two Numbers<\/h3>\r\nWhen multiplying two numbers,\r\n<ul id=\"eip-id1170322988632\">\r\n \t<li>if their signs are the same, the product is positive.<\/li>\r\n \t<li>if their signs are different, the product is negative.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiply decimal numbers.<\/h3>\r\n<ol id=\"eip-id1168469803517\" class=\"stepwise\">\r\n \t<li>Determine the sign of the product.<\/li>\r\n \t<li>Write the numbers in vertical format, lining up the numbers on the right.<\/li>\r\n \t<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\r\n \t<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\r\n \t<li>Write the product with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168466119810\" class=\"unnumbered unstyled\" summary=\"The top line says 3.9 times 4.075. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product. The signs are the same.<\/td>\r\n<td>The product will be positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the numbers in vertical format, lining up the numbers on the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221549\/CNX_BMath_Figure_05_02_019_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Place the decimal point. Add the number of decimal places in the factors [latex]\\left(1+3\\right)[\/latex]. Place the decimal point 4 places from the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is positive.<\/td>\r\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)=15.8925[\/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[ohm_question]146594[\/ohm_question]\r\n\r\n[ohm_question]146596[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(-8.2\\right)\\text{(}5.19\\text{)}[\/latex]\r\n[reveal-answer q=\"577733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"577733\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469608226\" class=\"unnumbered unstyled\" summary=\"The first line says negative 8.2 times 5.19. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(-8.2\\right)\\left(5.19\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are different.<\/td>\r\n<td>The product will be negative.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in vertical format, lining up the numbers on the right.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\\\ \\hfill 1038\\\\ \\underset{\\text{_____}}{4152}\\\\ \\hfill 42558\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221551\/CNX_BMath_Figure_05_02_020_img-01.png\" alt=\".\" \/><\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\\\ \\hfill 1038\\\\ \\underset{\\text{_____}}{4152}\\\\ \\hfill 42.558\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is negative.<\/td>\r\n<td>[latex]\\left(-8.2\\right)\\left(5.19\\right)=-42.558[\/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[ohm_question]146597[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to multiply two decimals.\r\n\r\nhttps:\/\/youtu.be\/55OtS_Dil1Y\r\n\r\nIn the next example, we\u2019ll need to add several placeholder zeros to properly place the decimal point.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(0.03\\right)\\text{(}0.045\\text{)}[\/latex]\r\n[reveal-answer q=\"460736\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"460736\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469489639\" class=\"unnumbered unstyled\" summary=\"The first line says 0.03 times 0.045. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(0.03\\right)\\left(0.045\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write in vertical format, lining up the numbers on the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221553\/CNX_BMath_Figure_05_02_021_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221554\/CNX_BMath_Figure_05_02_021_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221555\/CNX_BMath_Figure_05_02_021_img-01.png\" alt=\".\" \/>\r\n\r\nAdd zeros as needed to get the [latex]5[\/latex] places.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221556\/CNX_BMath_Figure_05_02_021_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is positive.<\/td>\r\n<td>[latex]\\left(0.03\\right)\\left(0.045\\right)=0.00135[\/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[ohm_question]146598[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Multiply by Powers of [latex]10[\/latex]<\/h2>\r\nIn many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[\/latex]. Let\u2019s see what happens when we multiply [latex]1.9436[\/latex] by some powers of [latex]10[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221557\/CNX_BMath_Figure_05_02_022_img.png\" alt=\"The top row says 1.9436 times 10, then 1.9436 times 100, then 1.9436 times 1000. Below each is a vertical multiplication problem. These show that 1.9436 times 10 is 19.4360, 1.9436 times 100 is 194.3600, and 1.9436 times 1000 is 1943.6000.\" \/>\r\nLook at the results without the final zeros. Do you notice a pattern?\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}1.9436\\left(10\\right)\\hfill &amp; =&amp; 19.436\\hfill \\\\ 1.9436\\left(100\\right)\\hfill &amp; =&amp; 194.36\\hfill \\\\ 1.9436\\left(1000\\right)\\hfill &amp; =&amp; 1943.6\\hfill \\end{array}[\/latex]<\/p>\r\nThe number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below\u00a0summarizes the results.\r\n<table id=\"fs-id3415452\" summary=\"A table is shown with five rows and three columns. The first row, which is the header row, reads 'Multiply by' in the first column, 'Number of zeros' in the second column, and 'Number of places decimal point moves' in the third column. The first column shows 10, 100, 1,000, and 10,000. The second row shows 1, 2, 3, 4. The third row shows '1 place to the right,' '2 places to the right,' '3 places to the right,' and '4 places to the right'.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Multiply by<\/th>\r\n<th>Number of zeros<\/th>\r\n<th>Number of places decimal point moves<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1[\/latex] place to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]100[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]1,000[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]3[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]10,000[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of [latex]10[\/latex] and then move the decimal point that same of places to the right.\r\n\r\nSo, for example, to multiply [latex]45.86[\/latex] by [latex]100[\/latex], move the decimal point [latex]2[\/latex] places to the right.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221604\/CNX_BMath_Figure_05_02_002_img.png\" alt=\"45.86 times 100 is shown to equal 4586. There is an arrow from the decimal going over 2 places from after the 5 to after the 6.\" \/>\r\nSometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let\u2019s multiply [latex]2.4[\/latex] by [latex]100[\/latex]. We need to move the decimal point [latex]2[\/latex] places to the right. Since there is only one digit to the right of the decimal point, we must write a [latex]0[\/latex] in the hundredths 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\/24221605\/CNX_BMath_Figure_05_02_003_img.png\" alt=\"2.4 times 100 is shown to equal 240. There is an arrow from the decimal going over 2 places from after the 2 to after the 0.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiply a decimal by a power of [latex]10[\/latex]<\/h3>\r\n<ol id=\"eip-id1168467251192\" class=\"stepwise\">\r\n \t<li>Move the decimal point to the right the same number of places as the number of zeros in the power of [latex]10[\/latex].<\/li>\r\n \t<li>Write zeros at the end of the number as placeholders if needed.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply [latex]5.63[\/latex] by factors of\r\n\r\n1. [latex]10[\/latex]\r\n\r\n2. [latex]100[\/latex]\r\n\r\n3. [latex]1000[\/latex]\r\n[reveal-answer q=\"296737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"296737\"]\r\n\r\nSolution\r\nBy looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.\r\n<table id=\"eip-id1168466081784\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, 'There are 2 zeros in 100, so move the decimal point 2 places to the right.' The image shows 5.63 with an arrow from the decimal point to after a line that has been added after the 3. The last line shows 563.\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.3\\left(10\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There is [latex]1[\/latex] zero in [latex]10[\/latex], so move the decimal point [latex]1[\/latex] place to the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221607\/CNX_BMath_Figure_05_02_023a_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466121817\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5.63\\left(100\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]2[\/latex] zeros in [latex]100[\/latex], so move the decimal point [latex]2[\/latex] places to the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221608\/CNX_BMath_Figure_05_02_023b_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]563[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469496249\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 1000. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5.63\\left(1000\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]3[\/latex] zeros in [latex]1000[\/latex], so move the decimal point [latex]3[\/latex] places to the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221609\/CNX_BMath_Figure_05_02_023c_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A zero must be added at the end.<\/td>\r\n<td>[latex]5,630[\/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>Key Takeaways<\/h3>\r\n[ohm_question]146599[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to multiply a decimal by 10, 100, and 1000.\r\n\r\nhttps:\/\/youtu.be\/JFAwf01nPG8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply two decimals together<\/li>\n<li>Multiply a decimal by 10, 100, or 1000<\/li>\n<\/ul>\n<\/div>\n<p>Multiplying decimals is very much like multiplying whole numbers\u2014we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.<\/p>\n<p>Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.<\/p>\n<p>So let\u2019s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side below. Look for a pattern.<\/p>\n<table id=\"fs-id1620547\" summary=\"A table is shown with 3 columns and 5 rows. The first row is a header row. The first column is not labeled. The other columns are labeled\">\n<thead>\n<tr valign=\"top\">\n<th>A<\/th>\n<th>B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\left(0.3\\right)\\left(0.7\\right)[\/latex]<\/td>\n<td>[latex]\\left(0.2\\right)\\left(0.46\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Convert to fractions.<\/td>\n<td>[latex]\\left({\\Large\\frac{3}{10}}\\right)\\left({\\Large\\frac{7}{10}}\\right)[\/latex]<\/td>\n<td>[latex]\\left({\\Large\\frac{2}{10}}\\right)\\left({\\Large\\frac{46}{100}}\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Multiply.<\/td>\n<td>[latex]{\\Large\\frac{21}{100}}[\/latex]<\/td>\n<td>[latex]{\\Large\\frac{92}{1000}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Convert back to decimals.<\/td>\n<td>[latex]0.21[\/latex]<\/td>\n<td>[latex]0.092[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.<\/p>\n<p>How many decimal places would you expect for the product of [latex]\\left(0.01\\right)\\left(0.004\\right)?[\/latex] If you said &#8220;five&#8221;, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors\u2014in this case two plus three\u2014to get the number of decimal places in the product\u2014in this case five.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221548\/CNX_BMath_Figure_05_02_018_img.png\" alt=\"The top line says 0.01 times 0.004 equals 0.00004. Below the 0.01, it says 2 places. Below the 0.004, it says 3 places. Below the 0.00004, it says 5 places. The bottom line says 1 over 100 times 4 over 1000 equals 4 over 100,000.\" \/><br \/>\nOnce we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.<\/p>\n<p>The rules for multiplying positive and negative numbers apply to decimals, too, of course.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two Numbers<\/h3>\n<p>When multiplying two numbers,<\/p>\n<ul id=\"eip-id1170322988632\">\n<li>if their signs are the same, the product is positive.<\/li>\n<li>if their signs are different, the product is negative.<\/li>\n<\/ul>\n<\/div>\n<p>When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiply decimal numbers.<\/h3>\n<ol id=\"eip-id1168469803517\" class=\"stepwise\">\n<li>Determine the sign of the product.<\/li>\n<li>Write the numbers in vertical format, lining up the numbers on the right.<\/li>\n<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\n<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\n<li>Write the product with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466119810\" class=\"unnumbered unstyled\" summary=\"The top line says 3.9 times 4.075. The next line says,\">\n<tbody>\n<tr>\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product. The signs are the same.<\/td>\n<td>The product will be positive.<\/td>\n<\/tr>\n<tr>\n<td>Write the numbers in vertical format, lining up the numbers on the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221549\/CNX_BMath_Figure_05_02_019_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Place the decimal point. Add the number of decimal places in the factors [latex]\\left(1+3\\right)[\/latex]. Place the decimal point 4 places from the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The product is positive.<\/td>\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)=15.8925[\/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=\"ohm146594\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146594&theme=oea&iframe_resize_id=ohm146594&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146596\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146596&theme=oea&iframe_resize_id=ohm146596&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(-8.2\\right)\\text{(}5.19\\text{)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q577733\">Show Solution<\/span><\/p>\n<div id=\"q577733\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469608226\" class=\"unnumbered unstyled\" summary=\"The first line says negative 8.2 times 5.19. The next line says,\">\n<tbody>\n<tr>\n<td>[latex]\\left(-8.2\\right)\\left(5.19\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are different.<\/td>\n<td>The product will be negative.<\/td>\n<\/tr>\n<tr>\n<td>Write in vertical format, lining up the numbers on the right.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\\\ \\hfill 1038\\\\ \\underset{\\text{_____}}{4152}\\\\ \\hfill 42558\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221551\/CNX_BMath_Figure_05_02_020_img-01.png\" alt=\".\" \/><\/td>\n<td>[latex]\\begin{array}{c}\\hfill 5.19\\\\ \\hfill \\underset{\\text{_____}}{\\times 8.2}\\\\ \\hfill 1038\\\\ \\underset{\\text{_____}}{4152}\\\\ \\hfill 42.558\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product is negative.<\/td>\n<td>[latex]\\left(-8.2\\right)\\left(5.19\\right)=-42.558[\/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=\"ohm146597\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146597&theme=oea&iframe_resize_id=ohm146597&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to multiply two decimals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples 2: Multiplication of Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/55OtS_Dil1Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next example, we\u2019ll need to add several placeholder zeros to properly place the decimal point.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(0.03\\right)\\text{(}0.045\\text{)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q460736\">Show Solution<\/span><\/p>\n<div id=\"q460736\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469489639\" class=\"unnumbered unstyled\" summary=\"The first line says 0.03 times 0.045. The next line says,\">\n<tbody>\n<tr>\n<td>[latex]\\left(0.03\\right)\\left(0.045\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The product is positive.<\/td>\n<\/tr>\n<tr>\n<td>Write in vertical format, lining up the numbers on the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221553\/CNX_BMath_Figure_05_02_021_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221554\/CNX_BMath_Figure_05_02_021_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221555\/CNX_BMath_Figure_05_02_021_img-01.png\" alt=\".\" \/><\/p>\n<p>Add zeros as needed to get the [latex]5[\/latex] places.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221556\/CNX_BMath_Figure_05_02_021_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>The product is positive.<\/td>\n<td>[latex]\\left(0.03\\right)\\left(0.045\\right)=0.00135[\/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=\"ohm146598\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146598&theme=oea&iframe_resize_id=ohm146598&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Multiply by Powers of [latex]10[\/latex]<\/h2>\n<p>In many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[\/latex]. Let\u2019s see what happens when we multiply [latex]1.9436[\/latex] by some powers of [latex]10[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221557\/CNX_BMath_Figure_05_02_022_img.png\" alt=\"The top row says 1.9436 times 10, then 1.9436 times 100, then 1.9436 times 1000. Below each is a vertical multiplication problem. These show that 1.9436 times 10 is 19.4360, 1.9436 times 100 is 194.3600, and 1.9436 times 1000 is 1943.6000.\" \/><br \/>\nLook at the results without the final zeros. Do you notice a pattern?<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}1.9436\\left(10\\right)\\hfill & =& 19.436\\hfill \\\\ 1.9436\\left(100\\right)\\hfill & =& 194.36\\hfill \\\\ 1.9436\\left(1000\\right)\\hfill & =& 1943.6\\hfill \\end{array}[\/latex]<\/p>\n<p>The number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below\u00a0summarizes the results.<\/p>\n<table id=\"fs-id3415452\" summary=\"A table is shown with five rows and three columns. The first row, which is the header row, reads 'Multiply by' in the first column, 'Number of zeros' in the second column, and 'Number of places decimal point moves' in the third column. The first column shows 10, 100, 1,000, and 10,000. The second row shows 1, 2, 3, 4. The third row shows '1 place to the right,' '2 places to the right,' '3 places to the right,' and '4 places to the right'.\">\n<thead>\n<tr valign=\"top\">\n<th>Multiply by<\/th>\n<th>Number of zeros<\/th>\n<th>Number of places decimal point moves<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex] place to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]100[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex] places to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]1,000[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex] places to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]10,000[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex] places to the right<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of [latex]10[\/latex] and then move the decimal point that same of places to the right.<\/p>\n<p>So, for example, to multiply [latex]45.86[\/latex] by [latex]100[\/latex], move the decimal point [latex]2[\/latex] places to the right.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221604\/CNX_BMath_Figure_05_02_002_img.png\" alt=\"45.86 times 100 is shown to equal 4586. There is an arrow from the decimal going over 2 places from after the 5 to after the 6.\" \/><br \/>\nSometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let\u2019s multiply [latex]2.4[\/latex] by [latex]100[\/latex]. We need to move the decimal point [latex]2[\/latex] places to the right. Since there is only one digit to the right of the decimal point, we must write a [latex]0[\/latex] in the hundredths place.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221605\/CNX_BMath_Figure_05_02_003_img.png\" alt=\"2.4 times 100 is shown to equal 240. There is an arrow from the decimal going over 2 places from after the 2 to after the 0.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Multiply a decimal by a power of [latex]10[\/latex]<\/h3>\n<ol id=\"eip-id1168467251192\" class=\"stepwise\">\n<li>Move the decimal point to the right the same number of places as the number of zeros in the power of [latex]10[\/latex].<\/li>\n<li>Write zeros at the end of the number as placeholders if needed.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply [latex]5.63[\/latex] by factors of<\/p>\n<p>1. [latex]10[\/latex]<\/p>\n<p>2. [latex]100[\/latex]<\/p>\n<p>3. [latex]1000[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q296737\">Show Solution<\/span><\/p>\n<div id=\"q296737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nBy looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.<\/p>\n<table id=\"eip-id1168466081784\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, 'There are 2 zeros in 100, so move the decimal point 2 places to the right.' The image shows 5.63 with an arrow from the decimal point to after a line that has been added after the 3. The last line shows 563.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.3\\left(10\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There is [latex]1[\/latex] zero in [latex]10[\/latex], so move the decimal point [latex]1[\/latex] place to the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221607\/CNX_BMath_Figure_05_02_023a_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>[latex]56.3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466121817\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td>[latex]5.63\\left(100\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There are [latex]2[\/latex] zeros in [latex]100[\/latex], so move the decimal point [latex]2[\/latex] places to the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221608\/CNX_BMath_Figure_05_02_023b_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>[latex]563[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469496249\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 1000. The next line says,\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]5.63\\left(1000\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There are [latex]3[\/latex] zeros in [latex]1000[\/latex], so move the decimal point [latex]3[\/latex] places to the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221609\/CNX_BMath_Figure_05_02_023c_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>A zero must be added at the end.<\/td>\n<td>[latex]5,630[\/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>Key Takeaways<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146599\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146599&theme=oea&iframe_resize_id=ohm146599&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to multiply a decimal by 10, 100, and 1000.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Multiply Decimals by 10, 100, and 1000\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JFAwf01nPG8?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-3919\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Multiply Decimals by 10, 100, and 1000. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JFAwf01nPG8\">https:\/\/youtu.be\/JFAwf01nPG8<\/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 146599 , 146598, 146597, 146596,146594. <strong>Authored by<\/strong>: Lumen Learning. <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, Shared previously<\/div><ul class=\"citation-list\"><li><strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/55OtS_Dil1Y\">https:\/\/youtu.be\/55OtS_Dil1Y<\/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>Examples 2: Multiplication of Decimals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/55OtS_Dil1Y\">https:\/\/youtu.be\/55OtS_Dil1Y<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/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":25777,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"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\":\"\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/55OtS_Dil1Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiply Decimals by 10, 100, and 1000\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/JFAwf01nPG8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146599 , 146598, 146597, 146596,146594\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Examples 2: Multiplication of Decimals\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/55OtS_Dil1Y\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3919","chapter","type-chapter","status-web-only","hentry"],"part":377,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3919","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3919\/revisions"}],"predecessor-version":[{"id":3920,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3919\/revisions\/3920"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/377"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3919\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3919"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3919"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3919"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}