EXAMPLE 1 Use properties of exponents Use the properties of rational exponents to simplify the expression. a. 7 1/ 7 1/2 7 (1/ + 1/2) 7 / b. (6 1/2 1/ ) 2 (6 1/2 ) 2 ( 1/ ) 2 6( 1/2 2 ) ( 1/ 2 ) 6 1 2/ 6 2/ c. ( ) 1/ [( ) ] 1/ 12 [ ( 1/)] (12 ) 1/ 12 1 1 12 d. 1 (1 1/) 2/ 1/ 1/ 2 e. 1/ 2 1/ 2 2 (7 1/ ) 2 7( 1/ 2 ) 7 2/ 6 1/ 6
EXAMPLE 2 Apply properties of exponents Biology A mammal s surface area S (in square centimeters) can be approximated by the model S km 2/ where m is the mass (in grams) of the mammal and k is a constant. The values of k for some mammals are shown below. Approximate the surface area of a rabbit that has a mass of. kilograms (. 10 grams).
EXAMPLE 2 SOLUTION Apply properties of exponents S km 2/ Write model. 9.7(. 10 ) 2/ Substitute 9.7 for k and. 10 for m. 9.7(.) 2/ (10 ) 2/ Power of a product property. 9.7(2.26)(10 2 ) Power of a product property. 2200 Simplify. ANSWER The rabbit s surface area is about 2200 square centimeters.
GUIDED PRACTICE for Examples 1 and 2 Use the properties of rational exponents to simplify the expression. 1. ( 1/ 7 1/ ) ( 1/ ) (7 1/ ) 2. 2 / 2 1/2. 2 (/ + 1/2) 2 / (1 1/) 1 / 1/ 1/ 1/ 7 1/ 1 7 / 7 /. 20 1/2 1/2 20 1/2 ( 1/2 ) (2 2 ) /2 8
GUIDED PRACTICE for Examples 1 and 2 Biology. Use the information in Example 2 to approximate the surface area of a sheep that has a mass of 9 kilograms (9. 10 grams). SOLUTION S km 2/ Write model. 9.7(9. 10 ) 2/ Substitute 9.7 for k and 9. 10 for m. 9.7(9.) 2/ (10 ) 2/ Power of a product property.
GUIDED PRACTICE for Examples 1 and 2 9.7(9) 10/ Power of a product property. 17,00 Simplify. ANSWER The Sheep s surface area is about 2200 square centimeters.
EXAMPLE Use properties of radicals Use the properties of radicals to simplify the expression. a. 12 18 12 8 216 6 Product property b. 80 80 16 2 Quotient property
EXAMPLE Write radicals in simplest form Write the expression in simplest form. a. 1 27 Factor out perfect cube. 27 Product property Simplify.
EXAMPLE Write radicals in simplest form b. 7 8 7 8 28 2 28 Simplify. 2 Make denominator a perfect fifth power. Product property
EXAMPLE Add and subtract like radicals and roots Simplify the expression. a. 10 + 7 10 (1 + 7) 10 8 10 b. 2 (8 1/ ) + 10 (8 1/ ) (2 +10) (8 1/ ) 12 (8 1/ ) c. 2 27 2 2 2 2 ( 1) 2 2 2
GUIDED PRACTICE for Examples,, and Simplify the expression. 6. 27 SOLUTION 27 27 81 Product property
GUIDED PRACTICE for Examples,, and 7. 20 2 SOLUTION 20 2 2 2 Factor out numerator to perfect cube. 2 2 Product property Simplify.
GUIDED PRACTICE for Examples,, and 8. SOLUTION 2 2 2 Simplify. 2 8 8 Make denominator a perfect fifth power. Product property
GUIDED PRACTICE for Examples,, and 9. + 0 SOLUTION + 0 + 2 + 2 (1+ 2)
EXAMPLE 6 Simplify expressions involving variables Simplify the expression. Assume all variables are positive. a. 6y 6 (y 2 ) (y 2 ) y 2 b. (27p q 12 ) 1/ 27 1/ (p ) 1/ (q 12 ) 1/ p ( 1/) q (12 1/) pq c. m n 8 m n 8 m m (n 2 ) n 2 d. 1xy 1/ 2x / z 6 7x (1 /) y 1/ z ( 6) 7x 1/ y 1/ z 6
EXAMPLE 7 Write variable expressions in simplest form Write the expression in simplest form. Assume all variables are positive. a. a 8 b 1 c a a b 10 b c a b 10 c a b Factor out perfect fifth powers. Product property ab 2 c a b Simplify. b. x y 8 x y 8 y y Make denominator a perfect cube. x y y 9 Simplify.
EXAMPLE 7 Write variable expressions in simplest form x y y 9 x y y Quotient property Simplify.
EXAMPLE 8 Add and subtract expressions involving variables Perform the indicated operation. Assume all variables are positive. a. 1 w + w 1 + w w b. xy 1/ 8xy 1/ ( 8) xy 1/ xy 1/ c. 12 2z z z 2 12z 2z 2 z 2z 2 (12z z) 2z 2 9z 2z 2
GUIDED PRACTICE for Examples 6, 7, and 8 Simplify the expression. Assume all variables are positive. 10. 27q 9 SOLUTION 27q 9 (q ) (q ) q 11. x 10 y SOLUTION x 10 y x 10 y (x 2 ) x2 y y
GUIDED PRACTICE for Examples 6, 7, and 8 12. 6xy / x 1/2 y 1/2 SOLUTION 6xy / x 1/2 y 1/2 2x (1 1/2) y (/ 1/2) 2x 1/2 y 1/ 1. 9w w w SOLUTION 9w w w 2 w 2 w 2 w w w 2 w w 2 w w 2 w 2w 2 w