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Algebra 1 Notes SOL A.3 (11.2) Radicals Mrs. Grieser Name: _______________________________ Block: _______ Date: _____________ Radical Review y means the xth root of y. The symbol is called a radical. x When x is not specified, we take the 2nd root, or the square root, of the number inside the radical (the radicand). A square root is a number multiplied by itself (squared) that gives us the radicand. Example: 4 = 2 A perfect square is a number whose square root is an integer. We can find other roots. To find the 3rd root, or the cube root, we want to find a number that when used as a factor 3 times (cubed) gives us the radicand. Example: 3 8 = 2 Calculators can help us find roots of numbers. We can also estimate roots. o Example: estimate 85 We know 85 lies between 81 and 100 , but is closer to 85 as being a little more than 9. 81 , so we can estimate Simplifying Radicals A radical expression is in simplest form if: No perfect squares other than 1 are in the radicand No fractions are in the radicand No radicals appear in the denominator of a fraction We use the Product Property of Radicals and Quotient Property of Radicals to simplify: Quotient Property of Radicals The square root of a quotient equals the quotient of the square roots of the a a numerator and denominator: b b Product Property of Radicals The square root of a product equals the product of the square roots of the factors: ab a b Write the radicand as the product of perfect squares, and then take the roots of those perfect squares. Examples for Product Property of Radicals: a) Simplify 32 = 32 16 2 = b) Simplify 16 ∙ 2 = 4 2 9x 3 = 9x 3 9 x2 x = You try: Simplify the radical expressions… a) Simplify 24 b) Simplify 1 25x 2 9 ∙ x 2 ∙ x = 3x x Algebra 1 Notes SOL A.3 (11.2) Radicals Mrs. Grieser Examples for Quotient Property of Radicals: a) Simplify 13 = 100 13 100 = 13 10 b) Simplify 1 = y2 1 y 2 = 1 y Multiplying Radicals We can use the product property of to multiply radicals. Examples: a) 6∙ 6 = 36 =6 (makes sense!) Note: When we solve b) 3x ∙4 x = 4 3x x = 4 3x = 4x 3 c) 7xy 2 ∙3 x = 3 7x 2 y 2 2 = 3xy 7 x 2 , we always take the positive value of x. Rationalizing Denominators By convention, an expression is not simplified if there is a radical expression in the denominator. The process of eliminating a radical in a denominator is called rationalizing the denominator. Examples: Rationalize the denominator… a) = = 5 7 5 7 b) ∙ 7 = 7 4 2x 4 2x 2x ∙ 2x = 4 2x 2 2x = 2x x 5 7 7 You try: Simplify the radical expressions… a) 20 d) 96x 2 g) 1 x b) 72 c) e) 9x 16 f) h) 2x 2 5 i) 2 32x 5 4 3 8 3n 2