# How to solve radical equations

This lesson will show how to solve radical equations. If an equation has a variable in the radicand or a variable with a fractional exponent, we call the equation a radical equation. Study the easy example below. If the index of the radical is 2, the equation is called square root equation.

## Solving square root equations

To solve square root equations, isolate the radical on one side of the equation. Then, raise both sides of the equation to the same power.

 $$Trick: \ If \ \sqrt[n]{y} = c, then \ (\sqrt[n]{y})^n = c^n \ and \ y = c^n$$
 $$4 + \sqrt{3x-2} = 6$$

Isolate the radical on one side of the equation

 $$4 - 4 + \sqrt{3x-2} = 6 - 4$$
 $$\sqrt{3x-2} = 2$$

Since you are dealing with a square root equation, raise both sides of the equation to the second power.

 $$(\sqrt{3x-2})^2 = 2^2$$

3x - 2 = 4

3x - 2 + 2 = 4 + 2

3x = 6

x = 2

## How to solve radical equations with rational exponents.

To solve equations with two radical expressions, isolate one of the radicals.

Solve (2x + 3)0.5 - (11x + 12)0.25 = 0

(2x + 3)0.5 - (11x + 12)0.25 + (11x + 12)0.25 = 0 + (11x + 12)0.25

(2x + 3)0.5 = (11x + 12)0.25

Raise both sides to the 4th power

[(2x + 3)0.5]4 = [(11x + 12)0.25]4

(2x + 3)2 = 11x + 12

4x2 + 12x + 9 = 11x + 12

4x2 + 12x - 11x + 9 - 12 = 11x - 11x + 12 - 12

4x2 + x - 3 = 0

(4x - 3)(x + 1) = 0

4x - 3 = 0

x = 3/4

x + 1 = 0

x = -1

x = -1 is a solution.

(2 times -1 + 3)0.5 - (11 times -1 + 12)0.25

10.5 - 10.25 = 1 - 1 = 0

x = 3/4 is also a solution.

(2 times 3/4 + 3)0.5 - (11 times 3/4 + 12)0.25

(6/4 + 3)0.5 - (33/4 + 12)0.25

(4.5)0.5 - (20.25)0.25

Is (4.5)0.5 - (20.25)0.25 = 0

Or is (4.5)0.5 = (20.25)0.25

Raise both sides to the 4th power

[(4.5)0.5]4 = [(20.25)0.25]4

(4.5)2 = 20.25

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