The n th root of a quotient is the quotient of the n th roots The n th root of

The n th root of a quotient is the quotient of the n

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The n th root of a quotient is the quotient of the n th roots. The n th root of a power is the power of the n th root. When n is odd, the n th root of a raised to the n th power is a. When n is even, the n th root of a raised to the n th power is the absolute value of a. 2 4 x 4 = ƒ x ƒ 2 3 x 3 = x 2 3 8 2 = A 1 3 8 B 2 = (2) 2 = 4 A 4 81 16 = 1 4 81 1 4 16 = 3 2 1 3 16 = 1 3 8 # 1 3 2 = 2 1 3 2
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66 CHAPTER 0 Prerequisites and Review EXAMPLE 4 Simplifying Radicals Simplify: a. b. Solution: a. b. since x 0 Ú 2 4 32 x 5 = 2 4 16 # 2 # x 4 # x = 1 4 16 # 1 4 2 # 2 4 x 4 # 1 4 x = 2 ƒ x ƒ 1 4 2 x = 2 x 1 4 2 x 2 3 - 24 x 5 = 2 3 ( - 8)(3) x 3 x 2 = 1 3 - 8 # 1 3 3 # 2 3 x 3 # 2 3 x 2 = - 2 x 2 3 3 x 2 2 4 32 x 5 2 3 - 24 x 5 - 2 s x s 2 s ƒ x ƒ s Combining Like Radicals We have already discussed properties for multiplying and dividing radicals. Now we focus on combining (adding or subtracting) radicals. Radicals with the same index and radicand are called like radicals . Only like radicals can be added or subtracted. EXAMPLE 5 Combining Like Radicals Combine the radicals if possible. a. b. c. d. Solution (a): Use the distributive property. Eliminate the parentheses. Solution (b): None of these radicals are alike. The expression is in simplified form. Solution (c): Write the radicands as products with a factor of 5. The square root of a product is the product of square roots. Simplify the square roots of perfect squares. All three radicals are now like radicals. Use the distributive property. Simplify. Solution (d): None of these radicals are alike because they have different indices. The expression is in simplified form. YOUR TURN Combine the radicals. a. b. 5 1 24 - 2 1 54 4 1 3 7 - 6 1 3 7 + 9 1 3 7 1 4 10 - 2 1 3 10 + 3 1 10 - 1 5 (3 + 2 - 6) 1 5 3 1 5 + 2 1 5 - 6 1 5 3 1 5 + 2 1 5 - 2(3) 1 5 3 1 5 + 1 4 # 1 5 - 2 1 9 # 1 5 3 1 5 + 1 20 - 2 1 45 = 3 1 5 + 1 4 # 5 - 2 1 9 # 5 2 1 5 - 3 1 7 + 6 1 3 = 5 1 3 4 1 3 - 6 1 3 + 7 1 3 = (4 - 6 + 7) 1 3 1 4 10 - 2 1 3 10 + 3 1 10 3 1 5 + 1 20 - 2 1 45 2 1 5 - 3 1 7 + 6 1 3 4 1 3 - 6 1 3 + 7 1 3 6 s Answer: a. b. 4 1 6 7 1 3 7
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0.6 Rational Exponents and Radicals 67 s 1 Rationalizing Denominators When radicals appear in a quotient, it is customary to write the quotient with no radicals in the denominator. This process is called rationalizing the denominator and involves multiplying by an expression that will eliminate the radical in the denominator. For example, the expression contains a single radical in the denominator. In a case like this, multiply the numerator and denominator by an appropriate radical expression, so that the resulting denominator will be radical free: If the denominator contains a sum of the form , multiply both the numerator and the denominator by the conjugate of the denominator, , which uses the difference of two squares to eliminate the radical term. Similarly, if the denominator contains a difference of the form , multiply both the numerator and the denominator by the conjugate of the denominator, . For example, to rationalize , take the conjugate of the denominator, which is : In general we apply the difference of two squares: Notice that the product does not contain a radical. Therefore, to simplify the expression multiply the numerator and denominator by : The denominator now contains no radicals: A 1 a - 1 b B ( a - b ) 1 A 1 a + 1 b B # A 1 a - 1 b B A 1 a - 1 b B A 1 a - 1 b B 1 A 1 a + 1 b B A 1 a + 1 b B A 1 a - 1 b B = A 1 a B 2 - A 1 b
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