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
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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