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Intermediate Algebra Section 7.3 – Simplifying Radical Expressions Notice that 9 ⋅ 4 = 3⋅ 2 = 6 and 9 ⋅ 4 = 36 = 6 . This implies that 9 ⋅ 4 = 9 ⋅ 4 . This illustrates the product rule for radicals. Product Property for Radicals If a and n b are real numbers, then Example: n a ⋅ n b = n ab Use the Product Property to simplify. 11 ⋅ 10 a) b) We have a similar rule for quotients. Notice which implies 3 10 ⋅ 3 5 4 2 = and 9 3 4 2 = , 9 3 4 4 = . 9 9 Quotient Property for Radicals If n a and n b are real numbers and n b is not zero, then n a na . = b nb Section 7.3 – Simplifying Radical Expressions Example: a) 4 Use the Quotient Property to simplify. 16y 81x 4 y10 6 9x b) Sometimes we use the reverse of the rule to simplify. Example: a) 27 b) 9x 5 y 7 c) 4 162m6 Simplify. Assume that all variables can be any real number. page 2 Section 7.3 – Simplifying Radical Expressions d) 3 40y10 Example: Simplify. Assume that all variables are greater than or equal to zero. 125x 2 a) b) 3 −81a 4b5 c) 5 32z Example: a) page 3 12 Simplify. Assume that all variables are greater than or equal to zero. 6 ⋅ 21 Section 7.3 – Simplifying Radical Expressions 8 x3 ⋅ 6 x b) c) 2 12 xy 3 ⋅ 3 30 x 2 y Example: 48a 5 3a a) 3 b) 128x 3 3 2x Example: a) Simplify. Assume that all variables are greater than zero. 237 Simplify. page 4 Section 7.3 – Simplifying Radical Expressions b) 4 page 5 9 ⋅ 6 12 Example: The radius r of a sphere whose volume V is given by 3V r=3 . 4π a) Write the radius of a sphere whose volume is 9 cubic centimeter as a radical in simplified form. b) Write the radius of a sphere whose volume is 32π cubic centimeters as a radical in simplified form.