Multiplying Monomials (7-1) Objective: Multiply monomials. Simplify expressions involving monomials.

Monomials A monomial is a number, a variable, or the product of a number and one or more variables with nonegative integer exponents. It has only one term. An expression that involves division by a variable is not a monomial. The monomial 3x is an example of a linear expression since the exponent of x is 1. The monomial 2x 2 is a nonlinear expression since the exponent is a positive number other than 1.

Example 1 Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. a. 17 – c  No, this expression has two terms. b. 8f 2 g  Yes, this is a product of numbers and variables. c. ¾  Yes, this is a constant. d. 5 / t  No, there is a variable in the denominator.

Check Your Progress Choose the best answer for the following. Which expression is a monomial? Which expression is a monomial? A.x 5 B.3p – 1 C. 9x / y D. c / d

Powers Recall that an expression of the form x n is called a power and represents the result of multiplying x by itself n times. x is the base, and n is the exponent. The word power is also used sometimes to refer to the exponent exponent base = 3 ∙ 3 ∙ 3 ∙ 3 = 81

Product of Powers By applying the definition of a power, you can find the product of powers. Look for a pattern in the exponents. 2 2 ∙ ∙ ∙ ∙ 4 2 To multiply two powers that have the same base, add their exponents. For any real number a and any integers m and p, a m ∙ a p = a m+p. b 3 ∙ b 5 = b 3+5 g 4 ∙ g 6 = g 4+6 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 2 6 = 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 4 5 = b 8 = g 10

Example 2 Simplify each expression. a. (r 4 )(-12r 7 ) =(1 ∙ -12)(r 4 ∙ r 7 ) =-12r 11 b. (6cd 5 )(5c 5 d 2 ) =(6 ∙ 5)(c 1 ∙ c 5 )(d 5 ∙ d 2 ) =30c 6 d 7

Check Your Progress Choose the best answer for the following. A. Simplify (5x 2 )(4x 3 ). A.9x 5 B.20x 5 C.20x 6 D.9x 6 (5 ∙ 4)(x 2 ∙ x 3 )

Check Your Progress Choose the best answer for the following. B. Simplify 3xy 2 (-2x 2 y 3 ). A.6xy 5 B.-6x 2 y 6 C.1x 3 y 5 D.-6x 3 y 5 (3 ∙ -2)(x 1 ∙ x 2 )(y 2 ∙ y 3 )

Power of a Power We can use the Product of Powers Property to find the power of a power. In the following examples, look for a pattern in the exponents. (3 2 ) 4 (3 2 ) 4 (r 4 ) 3 (r 4 ) 3 To find the power of a power, multiply the exponents. For any real number a and any integers m and p, (a m ) p = a m∙p. (b 3 ) 5 = b 3∙5 (g 6 ) 7 = g 6∙7 = (3 2 )(3 2 )(3 2 )(3 2 )= = 3 8 = (r 4 )(r 4 )(r 4 )= r = r 12 = b 15 = g 42

Example 3 Simplify [(2 3 ) 3 ] 2. = 2 3∙3∙2 = 2 18 = 262,144

Check Your Progress Choose the best answer for the following. Simplify [(4 2 ) 2 ] 3. Simplify [(4 2 ) 2 ] 3. A.4 7 B.4 8 C.4 12 D ∙2∙3

Power of a Product We can use the Product of Powers Property and the Power of a Power Property to find the power of a product. In the following examples, look for a pattern in the exponents. (tw) 3 (tw) 3 (2yz 2 ) 3 (2yz 2 ) 3 To find the power of a product, find the power of each factor and multiply. For any real numbers a and b and any integer m, (ab) m = a m b m. (-2xy 3 ) 5 = (-2) 5 (x) 5 (y 3 ) 5 = (tw)(tw)(tw) = (t 1 ∙ t 1 ∙ t 1 )(w 1 ∙ w 1 ∙ w 1 ) = t 3 w 3 = (2) 3 (y) 3 (z 2 ) 3 = 8y 3 z 6 = -32x 5 y 15 = (2yz 2 )(2yz 2 )(2yz 2 )

Example 4 Express the volume of a cube with side length 5xyz as a monomial. V = s 3 V = s 3 V = (5xyz) 3 V = (5xyz) 3 V = 5 3 x 3 y 3 z 3 V = 5 3 x 3 y 3 z 3 V = 125x 3 y 3 z 3 V = 125x 3 y 3 z 3

Check Your Progress Choose the best answer for the following. Express the surface area of the cube as a monomial. Express the surface area of the cube as a monomial. A.8p 3 q 3 B.24p 2 q 2 C.6p 2 q 2 D.8p 2 q 2 SA = 6s 2 SA = 6(2pq) 2 SA = 6 ∙ 2 2 p 2 q 2 SA = 6 ∙ 4p 2 q 2

Simplify Expressions We can combine and use these properties to simplify expressions involving monomials. To simplify a monomial expression, write an equivalent expression in which: Each variable base appears exactly once. Each variable base appears exactly once. There are no powers of powers, and There are no powers of powers, and All fractions are in simplest form. All fractions are in simplest form.

Example 5 Simplify [(8g 3 h 4 ) 2 ] 2 (2gh 5 ) 4. = (8g 3 h 4 ) 4 (2gh 5 ) 4 = 8 4 (g 3 ) 4 (h 4 ) 4 ∙ 2 4 (g) 4 (h 5 ) 4 = (4096 ∙ 16)(g 12 ∙ g 4 )(h 16 ∙ h 20 ) = 65,536g 16 h 36

Check Your Progress Choose the best answer for the following. Simplify [(2c 2 d 3 ) 2 ] 3 (3c 5 d 2 ) 3. Simplify [(2c 2 d 3 ) 2 ] 3 (3c 5 d 2 ) 3. A.1728c 27 d 24 B.6c 7 d 5 C.24c 13 d 10 D.5c 7 d 21 = (2c 2 d 3 ) 6 ∙ (3c 5 d 2 ) 3 = 2 6 (c 2 ) 6 (d 3 ) 6 ∙ 3 3 (c 5 ) 3 (d 2 ) 3 = (64 ∙ 27)(c 12 ∙ c 15 )(d 18 ∙ d 6 )