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Radicals Radicals The symbol a is a radical. The positive number a is the radicand. We say the square root of a or root a. The square root of a number n is the number which, when squared is n. n = n 2 For example : 5 and -5 are both the square root of 25, 2 2 since 5 = 25 and (-5) = 25. √ 25 = 5 and -5 The symbols: indicates the positive root 5 2 = 25 - 25 = 5 indicates the negative root -5 2 = 25 then then - 25 = -5 indicates both roots 25 = 5 and -5 Some examples: 8 - 36 = – 6 4 = ±2 64 = Why can’t we find a square root for 2 36... in other words can b = -36? 62 = 36 (-6)2 = 36 6(-6) = -36 The square of a number can never be negative. Therefore, the square root of a negative number does not exist in the real numbers. Radical Rules: Product Rule-The square root of a product is equal to the the product of the square roots. ab = a • b Quotient Rule-The square root of a quotient is equal to the quotient of the two radicals. a b a = b Important Products: n 2 n • n n2 n is positive =n Radicals are simplified when: 1) the radicand has no perfect square factors 2) the denominator of a fraction is never under a radical. The product and quotient rules allow radical expressions to be simplified. #1 “Take out” perfect square factors Rewrite the radicand as a product of its factors; with the largest perfect square factor possible. Use the product rule to simplify the root of the perfect square as a rational number, leaving the other factor under the radical. #2 Rationalize the denominator: To create a fraction with a rational denominator, multiply both numerator and denominator by the the irrational number found in the denominator of the fraction. Examples Comparison with variables