Multiplication properties of Exponents 7-1 Notes for Algebra 1 Multiplication properties of Exponents

7.1 pg. 395 21-63o, 69-84(x3)

Monomial (only one term) A number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. An expression that involves division by a variable is not a monomial.

Constant A monomial that is a real number.

Example 1: Identify Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 2.) 8 𝑓 2 𝑔 3.) 3 4 4.) 5 𝑡

Example 1: Identify Monomials Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 No, it involves subtraction, so it has more than one term. 2.) 8 𝑓 2 𝑔 Yes, it involves the product of a number and two variables. 3.) 3 4 Yes, the expression is a constant. 4.) 5 𝑡 No, it has a variable in the denominator.

Product of Powers 𝑥 3 ∙ 𝑥 6 𝑥 3+6 𝑥 9 To multiply two powers that have the same base, add their exponents. 𝑥 3 ∙ 𝑥 6 𝑥 3+6 𝑥 9

Example 2: Product of Powers Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2

Example 2: Product of Powers Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2 −12 𝑟 11 30 𝑐 6 𝑑 7

Power of a Power To find the power of a power, multiply the exponents. 𝑥 3 6 𝑥 3∙6 𝑥 18

Example 3: Power of a Power Simplify 2 3 3 2

Example 3: Power of a Power Simplify 2 3 3 2 2 18

Power of a Product 2 𝑥 2 𝑦 5 𝑧 4 3 2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3 To find the power of a product, find the power of each factor and multiply. 2 𝑥 2 𝑦 5 𝑧 4 3 2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3 8𝑥 6 𝑦 15 𝑧 12

Example 4: Power of a Product Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial.

Example 4: Power of a Product Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial. 125𝑥 3 𝑦 3 𝑧 3

Simplify Expressions To simplify a monomial expression, write an equivalent expression in which: Each variables base appears exactly once, There are no powers of powers, and All fractions are in simplest form.

Example 5: Simplify Expressions

Example 5: Simplify Expressions