Key Concepts

• To find the prime factorization of a number, use the factor tree method to divide away small prime numbers until only prime factors remain. The product of those prime factors is the prime factorization of the number.
• To find the least common multiple (LCM) of a list of numbers, write each number as a product of its prime factors, select the largest instance of each prime that appears in any one number, then multiply the selections together to obtain the LCM.
• To add or subtract fractions, first rewrite each fraction as an equivalent fraction, all having the same denominator, then add or subtract the numerators and place the result over the common denominator.
  • [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]
• To multiply fractions, place the product of the numerators over the product of the denominator.
  • [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
• To divide fractions, multiply the first fraction by the reciprocal of the second
  • [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex].
• To simplify a fraction, rewrite the numerator and denominator as products of their prime factors, then cancel ratios of common factors until there are no more common factors between the top and the bottom.
• Division by zero is undefined.
• When applying the order of operations, first simplify inside grouping symbols, then evaluate exponents or radicals, then multiply or divide from left to right in the order each appears, and finally add or subtract from left to right in the order each appears.

Glossary

composite number a natural number that can be written as the product of other natural numbers. For example, the number 10 can be written as the product of 2 and 5.

denominator the bottom part of a fraction – the denominator in the fraction [latex]\Large\frac{2}{3}[/latex] is [latex]3[/latex]

evaluate to substitute a given value for a variable in an expression and then perform the indicated mathematical operation

exponent, exponential notation a superscript, called the exponent or power, over a number, called the base, that tells how many times to use the base as a factor

expression several terms connected together by the operation of addition or subtraction

factor a number being multiplied, for example, for  [latex]2 \cdot 5 = 10[/latex] , the numbers [latex]2[/latex] and [latex]5[/latex] are factors of [latex]10[/latex]

least common multiple, LCM Given two or more numbers, the least common multiple between them is the smallest number that each of the given numbers divides evenly into

like terms terms in which the variables match exactly (exponents included)

numerator the top part of a fraction – the numerator in the fraction [latex]\Large\frac{2}{3}[/latex] is [latex]2[/latex]

operations/operators multiplication, division, addition, subtraction

prime number a number that is divisible only by itself and 1

product the result of  multiplication

radical a symbol indicating the square root, cube root, etc. of a number

simplify, reduce to write a mathematical statement in smallest terms

term a single number or variable, or the product of a number and a variable, which may also contain one or more exponents or radicals