2. Warm Up Write in exponential form. 1. 6 · 6 · 6 · 6 · 6 2. 3 x · 3 x · 3 x · 3 x Simplify. 3. 3 4 4. (–3) 5 5. (2 4 ) 5 6. (4 2 ) 0 6 5 (3 x ) 4 81 – 243 2 20 1
3. AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results in a monomial with an integer exponent. Also covered: AF1.3 California Standards
4. A monomial is a number or a product of numbers and variables with exponents that are whole numbers. To multiply two monomials, multiply the coefficients and add the exponents that have the same base. 7 x 5 , 3 a 2 b 3 , n 2 , 8, z 4 Not monomials Monomials m 3 ,4 z 2.5 , 5 + y , , 2 x 8 w 3
5. Multiply. Additional Example 1: Multiplying Monomials A. (3 a 2 )(4 a 5 ) (3 ∙ 4)( a 2 ∙ a 5 ) 12 a 7 Multiply coefficients. Add exponents that have the same base. B. (4 x 2 y 3 )(5 xy 5 ) (4 ∙ 5)( x 2 ∙ x )( y 3 ∙ y 5 ) Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 3 ∙ 4 ∙ a 2 + 5 (4 ∙ 5)( x 2 ∙ x 1 )( y 3 ∙ y 5 ) 4 ∙ 5 ∙ x 2 + 1 ∙ y 3+5 20 x 3 y 8 Use the Comm. and Assoc. Properties. Think: x = x 1 .
6. Multiply. Additional Example 1: Multiplying Monomials C. (–3 p 2 r )(6 pr 3 s) (–3 ∙ 6)( p 2 ∙ p )( r ∙ r 3 )( s ) Multiply coefficients. Add exponents that have the same base. – 3 ∙ 6 ∙ p 2 + 1 ∙ r 1+3 ∙ s – 18 p 3 r 4 s Use the Comm. and Assoc. Properties. (–3 ∙ 6)( p 2 ∙ p 1 )( r 1 ∙ r 3 )( s )
7. Multiply. Check It Out! Example 1 A. (2 b 2 )(7 b 4 ) (2 ∙ 7)( b 2 ∙ b 4 ) 14 b 6 Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 2 ∙ 7 ∙ b 2 + 4 B. (4 n 4 )(5 n 3 )( p ) (4 ∙ 5)( n 4 ∙ n 3 )( p ) 20 n 7 p Multiply coefficients. Add exponents that have the same base. Use the Comm. and Assoc. Properties. 4 ∙ 5 ∙ n 4 + 3 ∙ p
8. Multiply. Check It Out! Example 1 C. (–2 a 4 b 4 )(3 ab 3 c ) (–2 ∙ 3)( a 4 ∙ a )( b 4 ∙ b 3 )( c ) Multiply coefficients. Add exponents that have the same base. – 2 ∙ 3 ∙ a 4 + 1 ∙ b 4+3 ∙ c – 6 a 5 b 7 c Use the Comm. and Assoc. Properties. (–2 ∙ 3)( a 4 ∙ a 1 )( b 4 ∙ b 3 )( c )
9. To divide a monomial by a monomial, divide the coefficients and subtract the exponents of the powers in the denominator from the exponents of the powers in the numerator that have the same base.
10. Divide. Assume that no denominator equals zero. A. Divide coefficients. Subtract exponents that have the same base. Additional Example 2: Dividing Monomials 15 m 5 3 m 2 5 m 3 B. Divide coefficients. Subtract exponents that have the same base. 18 a 2 b 3 16 ab 3 m 5-2 15 3 a 2-1 b 3-3 9 8 a 9 8
11. Divide. Assume that no denominator equals zero. A. Divide coefficients. Subtract exponents that have the same base. Check It Out! Example 2 18 x 7 6 x 2 3 x 5 B. Divide coefficients. Subtract exponents that have the same base. 12 m 2 n 3 9 mn 2 x 7-2 18 6 m 2-1 n 3-2 4 3 mn 4 3
12. To raise a monomial to a power, you must first understand how to find a power of a product. Notice what happens to the exponents when you find a power of a product. ( xy ) 3 = xy ∙ xy ∙ xy = x ∙ x ∙ x ∙ y ∙ y ∙ y = x 3 y 3
13. Simplify. Additional Example 3: Raising a Monomial to a Power A. (3 y ) 3 3 3 ∙ y 3 27 y 3 Raise each factor to the power. B. (2 a 2 b 6 ) 4 2 4 ∙ ( a 2 ) 4 ∙ ( b 6 ) 4 16 a 8 b 24 Raise each factor to the power. Multiply exponents.
14. Simplify. Check It Out! Example 3 A. (4 a ) 4 4 4 ∙ a 4 256 a 4 Raise each factor to the power. B. (–3 x 2 y ) 2 (–3) 2 ∙ ( x 2 ) 2 ∙ ( y ) 2 9 x 4 y 2 Raise each factor to the power. Multiply exponents.
15. Lesson Quiz Multiply. 1. (3 g 2 h 3 )(–6 g 7 h 2 ) 2. (12 m 3 )(3 mp 3 ) Divide. Assume that no denominator equals zero. 3. 4. 5. 36 m 4 p 3 – 18 g 9 h 5 2 a 4 b 3 3 2 x – 5 p 3 6 a 6 b 4 3 a 2 b 9 x 3 y 6 x 2 y 20 p 5 q –4 p 2 q Simplify. 6. (–5 y 7 ) 3 7. (3 c 2 d 3 ) 4 8. (3 m 2 n ) 5 81 c 8 d 12 – 125 y 21 243 m 10 n 5
