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EXAMPLE 1

5 1. 5. 5 1/3. 5 1/3. a. 7 1/4 7 1/2. b. (6 1/2 4 1/3 ) 2. = (6 1/2 ) 2 (4 1/3 ) 2. = 6( 1/2 2 ) 4( 1/3 2 ). = 6 4 2/3. = 6 1 4 2/3. 1. c. (4 5 3 5 ) –1/5. = [(4 3) 5 ] –1/5. = 12 [5 (–1/5)]. =. 12. d. =. 2. 42. 42 1/3 2. 1/3. e. =. = 7( 1/3 2 ).

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EXAMPLE 1

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  1. 51 5 51/3 51/3 a. 71/4 71/2 b. (61/2 41/3)2 = (61/2)2 (41/3)2 = 6(1/22) 4(1/32) = 6 42/3 = 61 42/3 1 c. (45 35)–1/5 = [(4 3)5]–1/5 = 12[5 (–1/5)] = 12 d. = 2 42 421/3 2 1/3 e. = = 7(1/3 2) 6 61/3 EXAMPLE 1 Use properties of exponents Use the properties of rational exponents to simplify the expression. = 73/4 = 7(1/4 + 1/2) = 12 –1 = (125)–1/5 = 5(1 – 1/3) = 52/3 = 72/3 = (71/3)2

  2. A mammal’s surface area S(in square centimeters) can be approximated by the model S = km2/3 where m is the mass (in grams) of the mammal and kis a constant. The values of kfor some mammals are shown below. Approximate the surface area of a rabbit that has a mass of 3.4kilograms (3.4 103grams). EXAMPLE 2 Apply properties of exponents Biology

  3. km2/3 S = =9.75(3.4 103)2/3 Substitute 9.75 for kand 3.4 103 for m. 9.75(2.26)(102) 2200 ANSWER The rabbit’s surface area is about 2200square centimeters. EXAMPLE 2 Apply properties of exponents SOLUTION Write model. = 9.75(3.4)2/3(103)2/3 Power of a product property. Power of a product property. Simplify.

  4. a. 5 80 12 16 18 12 8 216 = = = 6 3 4 4 4 3 3 3 80 b. = = = 2 4 5 EXAMPLE 3 Use properties of radicals Use the properties of radicals to simplify the expression. Product property Quotient property

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

  6.      7 7 8 4 8 4 5 5 5 5 5 5    5 5 5 28 28 32 = b. = = 2 EXAMPLE 4 Write radicals in simplest form Make denominator a perfect fifth power. Product property Simplify.

  7.     2 2 2 2 2 3 3 3 3 3 (1 + 7) a. 8 7 + = = 4 3 4 4 4 3       3 10 10 54 27 10 10 b. = = + (81/5) 2 – 3 (3 – 1)  – – c.  2 3 = = = 2 (81/5) (81/5) 12 10 2 = (2 +10) (81/5) EXAMPLE 5 Add and subtract like radicals and roots Simplify the expression.

  8. 3  43(y2)3 a. 4y2 = = = 3pq4 b. (27p3q12)1/3 271/3(p3)1/3(q12)1/3 = = = 3 4 4 4 3p(3 1/3)q(12 1/3)     n8 43 m4 m4 m 3 3   (y2)3 64y6 14xy 1/3 4 7x1/4y1/3z6 7x(1 – 3/4)y1/3z –(–6) = = c. = = = n2 2x 3/4 z –6 4  (n2)4 d. m4 n8 EXAMPLE 6 Simplify expressions involving variables Simplify the expression. Assume all variables are positive.

  9. 5  = 5 4a8b14c5  4a5a3b10b4c5 5 5   a5b10c5 4a3b4 a. = = b. = x 5 ab2c  4a3b4 y8 x y 3 x y 3 = 3 y9 y8 y EXAMPLE 7 Write variable expressions in simplest form Write the expression in simplest form. Assume all variables are positive. Factor out perfect fifth powers. Product property Simplify. Make denominator a perfect cube. Simplify.

  10. 3  xy = 3  x y 3 =  y9 y3 EXAMPLE 7 Write variable expressions in simplest form Quotient property Simplify.

  11. 3z) (12z – 1 3 w w   + + 5 5 a. = = 9z 3 3   2z2 2z5 3 1 4 (3 – 8) xy1/4 – 5xy1/4 b. = = – 3xy1/4 8xy1/4 5 5 5 c. = z = 12 – w w   = 3 3 3 3 3z     – 2z2 2z2 54z2 2z2 12z EXAMPLE 8 Add and subtract expressions involving variables Perform the indicated operation. Assume all variables are positive.

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