EXAMPLE 3 Use properties of radicals Use the

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EXAMPLE 3 Use properties of radicals Use the properties of radicals to simplify the

EXAMPLE 3 Use properties of radicals Use the properties of radicals to simplify the expression. 3 a. b. 4 4 12 80 5 3 18 = = 4 3 12 18 = 3 216 = 6 80 = 4 16 = 2 5 Product property Quotient property

EXAMPLE 4 Write radicals in simplest form Write the expression in simplest form. a.

EXAMPLE 4 Write radicals in simplest form Write the expression in simplest form. a. 3 3 135 = 27 5 3 = 27 = 3 3 5 3 5 Factor out perfect cube. Product property Simplify.

EXAMPLE 4 5 b. 7 5 8 Write radicals in simplest form 5 =

EXAMPLE 4 5 b. 7 5 8 Write radicals in simplest form 5 = 7 5 8 5 4 5 4 Make denominator a perfect fifth power. 5 5 = 28 5 Product property 32 5 28 = 2 Simplify. 2 = 32 35 = 243 45 = 1024

EXAMPLE 5 Add and subtract like radicals and roots Indices must match! In this

EXAMPLE 5 Add and subtract like radicals and roots Indices must match! In this case each index is 4, so operation is allowed Simplify the expression. 4 4 a. 10 + 7 10 = (1 + 7) 10 = 8 10 b. 2 (81/5) + 10 (81/5) = (2 +10) (81/5) = 12 (81/5) c. 3 3 3 3 3 54 – 2 = 27 2 – 2 = 3 2 – 2 = (3 – 1) 2 = 2 2

GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. 6. 4 27

GUIDED PRACTICE for Examples 3, 4, and 5 Simplify the expression. 6. 4 27 4 3 8. 5 3 4 5 SOLUTION 7. 3 3 250 9. 3 24 2 SOLUTION 3 5 3 + 40 2 SOLUTION 5 SOLUTION 3 3 5

EXAMPLE 6 Simplify expressions involving variables Simplify the expression. Assume all variables are positive.

EXAMPLE 6 Simplify expressions involving variables Simplify the expression. Assume all variables are positive. 3 3 a. 3 b. (27 p 3 q 12)1/3 = 271/3(p 3)1/3(q 12)1/3 = 3 p(3 c. 4 64 y 6 = 43(y 2)3 4 = 43 3 (y 2)3 = 4 y 2 1/3)q(12 4 m m 4 = 2 = 4 8 n 2 4 n (n ) n 8 14 xy 1/3 1/4 y 1/3 z 6 (1 – 3/4)y 1/3 z –(– 6) 7 x 7 x d. = = 2 x 3/4 z – 6 1/3) = 3 pq 4

EXAMPLE 7 Write variable expressions in simplest form Write the expression in simplest form.

EXAMPLE 7 Write variable expressions in simplest form Write the expression in simplest form. Assume all variables are positive. a. 5 4 a 8 b 14 c 5 5 = 4 a 5 a 3 b 10 b 4 c 5 = = 5 a 5 b 10 c 5 ab 2 c 5 5 4 a 3 b 4 Factor out perfect fifth powers. Product property Simplify.

EXAMPLE 7 b. 3 x y 8 Write variable expressions in simplest form =

EXAMPLE 7 b. 3 x y 8 Write variable expressions in simplest form = x y y 8 y 3 = xy y 9 3 Make denominator a perfect cube. Simplify. 3 = x y 3 y 9 Quotient property 3 x y = 3 y Simplify.

EXAMPLE 8 Add and subtract expressions involving variables Perform the indicated operation. Assume all

EXAMPLE 8 Add and subtract expressions involving variables Perform the indicated operation. Assume all variables are positive. a. 1 3 1 + 3 4 w w 5 + 5 = 5 5 w = 5 w b. 3 xy 1/4 – 8 xy 1/4 = (3 – 8) xy 1/4 = – 5 xy 1/4 c. 3 12 2 z 5 –z 3 54 z 2 3 3 = 12 z 2 z 2 – 3 z 2 z 2 3 = (12 z – 3 z) 2 z 2 3 = 9 z 2 z 2

GUIDED PRACTICE for Examples 6, 7, and 8 Simplify the expression. Assume all variables

GUIDED PRACTICE for Examples 6, 7, and 8 Simplify the expression. Assume all variables are positive. 3 10. 27 q 9 SOLUTION 11. 5 12. 3 q 3 x 10 y 5 SOLUTION 6 xy 3/4 3 x 1/2 y 1/2 SOLUTION 2 x 1/2 y 1/4 13. 9 w 5 – w w 3 x 2 y SOLUTION 2 w 2 w