Lesson Objectives
- Learn how to solve radical equations with higher-level roots
How to Solve Radical Equations with Higher-Level Roots
Let's begin by restating our procedure for solving an equation with radicals. We can then apply this procedure to equations with higher-level roots.
In this case, we will move the rightmost radical to the right side of the equation. This will isolate both radicals. $$\sqrt[3]{2x - 11}=\sqrt[3]{5x + 1}$$ Step 2) Raise both sides of the equation to a power equal to the index of the radical.
In this case, we have a cube root, this means we want to cube both sides of the equation. $$\left(\sqrt[3]{2x - 11}\right)^3=\left(\sqrt[3]{5x + 1}\right)^3$$ $$2x - 11=5x + 1$$ Step 3) Solve the equation. $$2x - 11=5x + 1$$ $$-3x=12$$ $$x=-4$$ Step 4) Check all solutions in the original equation. $$\sqrt[3]{2x - 11}- \sqrt[3]{5x + 1}=0$$ $$\sqrt[3]{2(-4) - 11}- \sqrt[3]{5(-4) + 1}=0$$ $$\sqrt[3]{-19}- \sqrt[3]{-19}=0$$ $$0=0 \hspace{.2em}\color{green}{✔}$$
Solving Equations with Radicals
- Isolate one of the radicals
- Raise both sides of the equation to a power equal to the index of the radical
- Repeat the previous two steps if necessary
- Solve the equation
- Check all solutions in the original equation
Solving Radical Equations with Higher-Level Roots
Example 1: Solve each equation $$\sqrt[3]{2x - 11}- \sqrt[3]{5x + 1}=0$$ Step 1) Isolate one of the radicals.In this case, we will move the rightmost radical to the right side of the equation. This will isolate both radicals. $$\sqrt[3]{2x - 11}=\sqrt[3]{5x + 1}$$ Step 2) Raise both sides of the equation to a power equal to the index of the radical.
In this case, we have a cube root, this means we want to cube both sides of the equation. $$\left(\sqrt[3]{2x - 11}\right)^3=\left(\sqrt[3]{5x + 1}\right)^3$$ $$2x - 11=5x + 1$$ Step 3) Solve the equation. $$2x - 11=5x + 1$$ $$-3x=12$$ $$x=-4$$ Step 4) Check all solutions in the original equation. $$\sqrt[3]{2x - 11}- \sqrt[3]{5x + 1}=0$$ $$\sqrt[3]{2(-4) - 11}- \sqrt[3]{5(-4) + 1}=0$$ $$\sqrt[3]{-19}- \sqrt[3]{-19}=0$$ $$0=0 \hspace{.2em}\color{green}{✔}$$
Skills Check:
Example #1
Solve each equation. $$\sqrt[3]{2x + 3}=5$$
Please choose the best answer.
A
$$x=-\frac{1}{8}, 7$$
B
$$x=-25, 13$$
C
$$x=-5$$
D
$$x=61$$
E
$$x=-1$$
Example #2
Solve each equation. $$\sqrt{2 \sqrt{x - 3}}=\sqrt{4 - x}$$
Please choose the best answer.
A
$$x=-13, 17$$
B
$$x=-\frac{1}{8}, 7$$
C
$$x=-3, 7$$
D
$$x=1 - 5 \sqrt{3}$$
E
$$x=6 - 2 \sqrt{2}$$
Example #3
Solve each equation. $$\sqrt[4]{2x - 9}=0$$
Please choose the best answer.
A
$$x=-\frac{1}{2}$$
B
$$x=\frac{9}{2}$$
C
$$x=-3, 7$$
D
$$x=-2, 1$$
E
$$x=-\frac{1}{4}, 9$$
Congrats, Your Score is 100%
Better Luck Next Time, Your Score is %
Try again?
Ready for more?
Watch the Step by Step Video Lesson Take the Practice Test
