A90 APPENDIX Review 4 Solve Radical Equations When the variable in an equation occurs in a square root, cube root, and so on—that is, when it occurs in a radical—the equation is called a radical equation. Sometimes a suitable operation will change a radical equation to one that is linear or quadratic. A commonly used procedure is to isolate the most complicated radical on one side of the equation and then eliminate it by raising both sides to a power equal to the index of the radical. Care must be taken, however, because apparent solutions that are not, in fact, solutions of the original equation may result. These are called extraneous solutions. Therefore, we need to check all answers when working with radical equations, and we check them in the original equation. CAUTION Do not distribute the h in the denominator. This allows us to divide out the h’s in the last step. j Rationalizing Numerators Rationalize the numerator of the following expression. x h x h h 0 + − ≠ Solution EXAMPLE 6 The numerator contains the factor x h x, + − so we multiply the numerator and denominator by x h x + + to obtain x h x h x h x h x h x x h x x h x h x h x x h x h x h x h h x h x x h x 1 2 2 ( ) ( ) ( ) ( ) ( ) + − = + − ⋅ + + + + = + − + + = + − + + = + + = + + Now Work problem 69 Now Work problem 77 Solving a Radical Equation Find the real solutions of the equation: x2 4 2 0 3 − − = Solution EXAMPLE 7 The equation contains a radical whose index is 3. Isolate it on the left side. x x 2 4 2 0 2 4 2 3 3 − − = − = Add 2 to both sides. Now raise both sides to the third power (the index of the radical is 3) and solve. x x x x 2 4 2 2 4 8 2 12 6 3 3 3 ( ) − = − = = = Raise both sides to the power 3. Simplify. Add 4 to both sides. Divide both sides by 2. Check: 26 42 1242 82 22 0 3 3 3 ( ) −−= −−= −=−= The solution set is 6 . { }
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