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6.2 Properties of Exponents

6.2 Properties of Exponents. 8 5. 2. 8 –4. Lesson 6.2 , For use with pages 420-427. Simplify the expression. 1. 4 3 • 4 8. 4 11. ANSWER. 8 9. ANSWER. 3. √48. 4√3 . 4. √30 • √15. 15√2 . Lesson 6.2 , For use with pages 420-427. Simplify the expression. ANSWER. ANSWER.

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6.2 Properties of Exponents

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  1. 6.2 Properties of Exponents

  2. 85 2. 8–4 Lesson 6.2, For use with pages 420-427 Simplify the expression. 1. 43 • 48 411 ANSWER 89 ANSWER

  3. 3. √48 4√3 4. √30 • √15 15√2 Lesson 6.2, For use with pages 420-427 Simplify the expression. ANSWER ANSWER

  4. 5. √80 – √20 2√5 Lesson 6.2, For use with pages 420-427 Simplify the expression. ANSWER

  5. 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

  6. 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

  7. 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.

  8. 3 31/4 2. 23/4 21/2 1. (51/3 71/4)3 5 73/4 25/4 33/4 3. 201/2 3 8 4. 51/2 for Examples 1 and 2 GUIDED PRACTICE Use the properties of rational exponents to simplify the expression. SOLUTION SOLUTION SOLUTION SOLUTION

  9. Use the information in Example 2 to approximate the surface area of a sheep that has a mass of 95kilograms(9.5 104grams). for Examples 1 and 2 GUIDED PRACTICE Biology 17,500cm2 SOLUTION

  10. 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

  11.  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.

  12.      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.

  13.     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.

  14. 3   5 3 24 2   3 250 + 40 3 5 4 4 4 27 3 3 3 3   5 5 3 5 2 for Examples 3, 4, and 5 GUIDED PRACTICE Simplify the expression. SOLUTION SOLUTION SOLUTION SOLUTION

  15. 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.

  16. 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.

  17. 3  xy = 3  x y 3 =  y9 y3 EXAMPLE 7 Write variable expressions in simplest form Quotient property Simplify.

  18. 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.

  19. 6xy 3/4 3x 1/2 y 1/2 3q3 3  27q9 5 2x1/2y1/4 –  w  w3 9w5 x10 y5 x2 y w 2w2  for Examples 6, 7, and 8 GUIDED PRACTICE Simplify the expression. Assume all variables are positive. SOLUTION SOLUTION SOLUTION SOLUTION

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