Copyright © Cengage Learning. All rights reserved. 8 Radical Functions Copyright © Cengage Learning. All rights reserved.

8.4 Solving Radical Equations Copyright © Cengage Learning. All rights reserved.

Objectives Solve radical equations involving one or more square roots. Use radical equations to solve application problems. Solve radical equations involving higher roots.

Solving Radical Equations

Solving Radical Equations A radical can be written using fraction exponents. Using the fraction exponent representation of radicals, we can solve radical equations using the same basic steps to solve power equations. In the case of a square root, isolating the square root and then squaring both sides will eliminate the radical and allow us to solve for the variable.

Solving Radical Equations

Example 1 – Solving radical equations involving one square root Solve the following equations. Check the answers in the original equation. Solution: a. The radical is already isolated on one side, so square both sides of the equation to eliminate the radical. Square both sides of the equation.

Example 1 – Solution cont’d x = 25 The square root of 25 is 5, so this is the correct answer. b. Start by adding 3 to both sides to isolate the radical on one side of the equation. Then we square both sides to eliminate the radical. Add 3 to both sides of the equation.

Example 1 – Solution x + 2 = 9 x = 7 cont’d x + 2 = 9 x = 7 Check this answer by substituting x = 7 into the original equation. 3 = 3 Square both sides of the equation. Solve the remaining equation by subtracting 2 from both sides. The answer works.

Solving Radical Equations Many applications in physics and other sciences use formulas that contain radicals. Often these formulas need to be solved for a variable that is inside the radical itself. In the next example, we will see such formulas.

Example 2 – Period of a pendulum Traditionally, clocks have used pendulums to keep time. The period (the time to go back and forth) of a simple pendulum for a small amplitude is given by the function where T (L) is the period in seconds and L is the length of the pendulum in feet. Use this formula to find the following: a. The period of a pendulum if its length is 2 feet. b. How long must a pendulum be if we want the period to be 3 seconds?

Example 2(a) – Solution Because the length of the pendulum is 2 feet, we know that L = 2, so we get T (2)  1.57 Therefore, a 2-foot pendulum will have a period of about 1.57 seconds.

Example 2(b) – Solution We are given that T (L) = 3, so we get cont’d Isolate the radical on one side of the equation.

Example 2(b) – Solution 7.295  L cont’d Square both sides of the equation to eliminate the radical. Solve for L.

Example 2(b) – Solution We can check this answer with the table. cont’d We can check this answer with the table. To have a period of 3 seconds, we will need a pendulum that is about 7.3 feet long.

Solving Radical Equations Involving More Than One Square Root

Solving Radical Equations Involving More Than One Square Root Some equations may have more than one radical involved and will require more work to solve. You will want to first isolate one of the radicals so that squaring both sides will eliminate that radical and then work on the remaining problem.

Example 4 – Solving radical equations with more than one square root Solve the following. Solution: a. x + 1 = 2 x Square both sides to eliminate the radical. Solve for x.

Example 4 – Solution b. cont’d Check the answer. The answer works. One radical is isolated. Square both sides to eliminate the isolated radical. The right side will need the distributive property because of the addition involved.

Example 4 – Solution cont’d Isolate the remaining radical.

Example 4 – Solution cont’d Square both sides again to eliminate the remaining radical. Check the answer. The answer works.

Example 4 – Solution c. cont’d Square both sides. Use the distributive property. Simplify.

Example 4 – Solution cont’d Isolate the radical. Square both sides.

Example 4 – Solution cont’d Check the answer. The answer works.

Solving Radical Equations Involving Higher Order-Roots

Solving Radical Equations Involving Higher-Order Roots When an equation has a higher-order root, the solving process will be basically the same as with square roots. After isolating the radical, raise both sides of the equation to the power of the roots index. If the root has an index of n, then we raise both sides of the equation to the nth power. Another way to think of this process is to consider the root as a fraction exponent.

Solving Radical Equations Involving Higher-Order Roots If an equation has a higher-order root we can undo the fraction exponent by raising both sides to the reciprocal exponent. x = 2187 Rewrite the radical using a fraction exponent. Raise both sides to the reciprocal exponent.

Solving Radical Equations Involving Higher-Order Roots

Example 7 – Solving higher-order radical equations Solve the following equations. a. b. c. Solution: a. 2x = 125 x = 62.5 Because this is a cube root, raise both sides to the third power.

Example 7 – Solution 5 = 5 b. 4 x – 9 = 32 4 x = 41 x = 10.25 cont’d Check the answer. The answer works. Because this is a fifth root, raise both sides to the power of 5.

Example 7 – Solution cont’d 2 = 2 c. Again, we will isolate the radical. Since this is a cube root, we will raise both sides to the power of 3, the reciprocal power. Check the answer. The answer works. Isolate the radical by dividing both sides of the equation by 5.

Example 7 – Solution x – 7 = 512 x = 519 40 = 40 cont’d Raise both sides of the equation to the reciprocal exponent, 3. Solve the remaining equation by adding 7 to both sides. Check the answer. The answer works.