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Radicals and Rational Exponents Chapter 10

10 Radicals and Rational Exponents 10.1 Finding Roots 10.3 Simplifying Expressions Containing Square Roots 10.4 Simplifying Expressions Containing Higher Roots 10.5 Adding, Subtracting, and Multiplying Radicals 10.6 Dividing Radicals Putting it All Together 10.7 Solving Radical Equations 10.8 Complex Numbers

10.7 Solving Radical Equations In this section, we will learn how to solve radical equations. An equation containing A variable in the radicand is a radical equation. Some examples are listed below. Understand the Steps for Solving a Radical Equation An extraneous solution is a value that satisfies one of the new equations but does not satisfy the original equation. Extraneous solutions occur frequently when solving radical equations, so we must check all possible solutions in the original equation and discard any that are extraneous.

Solve an Equation Containing One Square Root Example 1 Solve. Solution Step 1: The radical is on a side by itself: Step 2: Square both sides to eliminate the square root. Square both sides. Step 3 and 4: do not apply because there are no like terms to combine and no radicals remain. Step 5: Solve the equation. Add 8 to each side. Step 6: Check b = 12 in the original equation. The solution set is {12}.

The first step is to get the radical on a side by itself. Example 2 Solve. Solution The first step is to get the radical on a side by itself. Subtract 7 from each side. Square both sides to eliminate radical. The square root has been eliminated. Subtract 3 from each side. Check w = 13 in the original equation. Because w = 13 is an extraneous solution, The equation has no real solution. False

Start by getting the radical on a side by itself. Example 3 Solve. Solution Start by getting the radical on a side by itself. Add 2 to each side. Square both sides to eliminate the radical. Simplify; square the binomial. Subtract 2y; subtract 4.. Factor. or Set each factor equal to zero. or Solve. The solution set is {-2,0}. Check y =0 and y = -2 in the original equation.

Solve an Equation Containing Two Square Roots Example 4 Solve. Solution Begin by getting a radical on a side by itself. Add to each side. Square both sides to eliminate the radical. Distribute 4 into parenthesis. Solve equation. Check b = 8 in the original equation. The solution set is {8}.

Example 5 Square and simplify Solution Substitute 6 for a and for b. Multiply Combine like terms.

Example 6 Solve each equation. Solution Subtract on both sides. Divide by negative one on both sides. Square both sides. Simplify. Notice the equation still contains a radical. Therefore, repeat Steps 1-3. Begin by getting radical on a side by itself. Subtract 1 and c on both sides. Divide by -2 on both sides and square both sides. Simplify. Check is left to the student. The solution set is {16}.

1) Do not square both sides before getting a radical on a side by Watch out for two common mistakes that students make when solving an equation like the one in Example 5b. 1) Do not square both sides before getting a radical on a side by itself. This is incorrect: 2) The second time we perform step 2, watch out for this common error. On the left we must multiply using FOIL or the formula and on the right we must remember to square the 2. Be Careful

Solve an Equation Containing a Cube Root Example 7 Solve. Solution The steps that were used to solve equations containing square roots can be used to eliminate cube roots. Except to eliminate a cube root, we cube both sides of the equation. Subtract on both sides. Cube both sides to eliminate the radicals. Simplify : Distribute: Subtract 7a; add 8. Check a = 9 in the original equation. The solution set is {9}.