13 Exponents and Polynomials

13.2 The Product Rule and Power Rules for Exponents Objectives 1. Use exponents. 2. Use the product rule for exponents. 3. Use the rule (am)n = amn. 4. Use the rule (ab)m = ambm. 5. Use the rule (a/b)m = am/bm. 6. Use combinations of the rules for exponents. 7. Use the rules for exponents in a geometry application.

Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate. Use Exponents Example 1 Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate. Since 2 occurs as a factor 6 times, the base is 2 and the exponent is 6. The exponential expression is 26, read “2 to the sixth power” or simply “2 to the sixth.” Exponent (or Power) 2 = 2 · 2 · 2 · 2 · 2 · 2 6 = 64 Base 6 factors of 2

Example 2 Evaluate. Name the base and the exponent. Base Exponent Use Exponents Example 2 Evaluate. Name the base and the exponent. Base Exponent (a) 2 = 2 · 2 · 2 · 2 4 2 4 = 16 (b) – 2 4 4 = –1 · 2 = –1 · 2 · 2 · 2 · 2 2 4 = –16 (c) (–2) 4 = (–2) (–2) (–2) (–2) –2 4 = 16

In summary, and are not necessarily the same. Use Exponents CAUTION Expression Base Exponent Example – a n – 5 2 a n = – ( 5 · 5 ) = – 25 (–a) n (– 5) 2 (–a) n = (–5) (–5) = 25 – a n (–a) n In summary, and are not necessarily the same.

Use the Product Rule for Exponents a m · a n = a m + n For any positive integers m and n, (Keep the same base and add the exponents.) 5 3 · 5 4 Example: = 5 3 + 4 = 5 7 ( 5 · 5 · 5 ) ( 5 · 5 · 5 · 5 )

Use the Product Rule for Exponents CAUTION Do not multiply the bases when using the product rule. Keep the same base and add the exponents. Example:

Use the Product Rule for Exponents Example 3 Use the product rule for exponents to simplify, if possible. 5 8 (a) 7 · 7 = 7 5 + 8 = 7 13 4 5 (b) (–2) (–2) = (– 2) 4 + 5 = (– 2) 9 5 (c) y · y 5 = y · y 1 = y 1 + 5 = y 6

Use the Product Rule for Exponents Example 3 (concluded) Use the product rule for exponents to simplify, if possible. 2 4 (d) n · n = n 2 + 4 = n 6 2 (e) 3 · 2 The product rule does not apply because the bases are different. = 9 · 4 = 36 3 2 (f) 2 + 2 The product rule does not apply because it is a sum, not a product. = 8 + 4 = 12

Use the Product Rule of Exponents Example 4 Commutative and associative properties Multiply; product rule Add the exponents.

Use the Product Rule for Exponents CAUTION Be sure you understand the difference between adding and multiplying exponential expressions. For example, 2 7k + 3k 2 = ( 7 + 3 )k = 10k , Add. 2 7k 3k 4 2 + 2 = ( 7 · 3 )k = 21k . Multiply. but,

Power Rule (a) for Exponents For any positive integers m and n, Use the Rule (am)n = amn Power Rule (a) for Exponents For any positive integers m and n, m n ( a ) m n = a . (Raise a power to a power by multiplying exponents.) 3 2 ( 4 ) 3 · 2 = 4 6 = 4 . Example:

Use power rule (a) for exponents to simplify. Use the Rule (am)n = amn Example 5 Use power rule (a) for exponents to simplify. (a) ( 3 ) 2 5 = 3 2 · 5 = 3 10 (b) ( 4 ) 8 6 = 4 8 · 6 = 4 48 (c) ( n ) 7 3 = n 7 · 3 = n 21 (d) ( 2 ) 5 8 = 2 5 · 8 = 2 40

Use the Rule (ab)m = ambm Power Rule (b) for Exponents For any positive integer m, m ( ab ) m m = a b . (Raise a product to a power by raising each factor to the power.) 3 ( 5h ) 3 = 5 h . Example:

Use the Rule (ab)m = ambm Example 6 Use power rule (b) for exponents to simplify. (a) ( 2abc ) 4 = 2 a b c 4 Power rule (b) = 16 a b c 4 (b) 5 ( x y ) 2 3 = 5 ( x y ) 2 6 Power rule (b) = 5x y 2 6

Use the Rule (ab)m = ambm Example 6 (continued) Use power rule (b) for exponents to simplify. (c) 7 ( 2m n p ) 7 5 3 = 7 [ 2 ( m ) ( n ) ( p ) ] 3 5 7 1 Power rule (b) = 7 [ 8 m n p ] 3 15 21 Power rule (a) = 56 m n p 3 15 21

Use the Rule (ab)m = ambm Example 6 (concluded) Use power rule (b) for exponents to simplify. (d) ( –3 ) 5 4 = ( – 1 · 3 ) 5 4 – a = – 1 · a = ( – 1 ) ( 3 ) 5 4 Power rule (b) = – 1 · 3 20 Power rule (a) = – 3 20

Use the Rule (ab)m = ambm CAUTION Power rule (b) does not apply to a sum:

Use the Rule (a/b)m = am/bm Power Rule (c) for Exponents For any positive integer m, m a b = ( b ≠ 0 ). (Raise a quotient to a power by raising both the numerator and the denominator to the power.) Example: 2 3 4 =

Use the Rule (a/b)m = am/bm Example 7 Use power rule (c) for exponents to simplify. 4 2 5 4 2 5 = 16 625 = (a) 7 x y 7 x y = (b) y ≠ 0

For any positive integers m and n: Examples Rules for Exponents For any positive integers m and n: Examples a m · a n = a m + n 3 4 · 3 5 = 3 9 Product rule m n ( a ) m n = a (a) 4 5 ( 2 ) 20 = 2 Power rules m m m (b) ( ab ) = a b 3 ( 4k ) = 4 k (c) m a b = 3 4 7 = ( b ≠ 0 ).

Use Combinations of the Rules for Exponents Example 8 Simplify each expression. (a) · 3 5 2 3 4 = · 2 3 4 5 1 Power rule (c) = 2 3 · 4 · 5 3 1 Multiply fractions. = 3 4 2 7 Product rule

Use Combinations of the Rules for Exponents Example 8 (continued) Simplify each expression. (b) ( 7m n ) ( 7m n ) 2 3 5 = ( 7m n ) 2 8 Product rule = 7 m n 16 8 Power rule (b)

Use Combinations of the Rules for Exponents Example 8 (continued) Simplify each expression. (c) ( 2x y ) ( 2 x y ) 10 6 3 7 4 = ( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y ) 10 7 4 6 3 Power rule (b) = 2 · x · y · 2 · x · y 10 70 40 6 18 Power rule (a) = 2 · 2 · x · x · y · y 70 6 40 10 18 Commutative and associative properties = 2 · x · y 16 88 46 Product rule

Use Combinations of the Rules for Exponents Example 8 (concluded) Simplify each expression. (d) ( – g h ) ( – g h ) 3 2 5 = ( – 1 g h ) ( – 1 g h ) 3 2 5 = ( – 1 ) ( g ) ( h ) ( – 1 ) ( g ) ( h ) 2 6 3 15 Power rule (b) = ( – 1 ) ( g ) ( h ) 5 12 17 Product rule = – 1 g h 12 17

Use the Rules for Exponents in a Geometry Application Example 9 Find a polynomial that represents the area of the geometric figure. 4x 3 2x 2 (a) Use the formula for the area of a rectangle, A = LW. A = ( 4x )( 2x ) 3 2 A = 8x 5 Product rule

Use the Rules for Exponents in a Geometry Application Example 9 Find a polynomial that represents the area of the geometric figure. 8n 5 3n 4 1 2 (b) Use the formula for the area of a triangle, A = LW. A = ( 3n ) ( 8n ) 4 5 1 2 A = ( 24n ) 9 1 2 Product rule A = 12n 9