Radical Expressions in many equations must be simplified in a special way, which we will teach you how to simplify them in this blog post.

## Related Topics

- How to Rationalize Radical Expressions
- How to Solve Radical Equations
- How to Multiply Radical Expressions
- How to Rationalize Radical Expressions
- How to Find Domain and Range of Radical Functions

## A step-by-step guide to simplifying radical expressions

- Find the prime factors of the numbers or expressions inside the radical.
- Use radical properties to simplify the radical expression:

\(\sqrt[n]{x^a }=x^{\frac{a}{n}}, \sqrt[n]{xy }=x^{\frac{1}{n}}×y^{\frac{1}{n}}, \sqrt[n]{\frac{x}{y} }=\frac{x^{\frac{1}{n}}}{y^{\frac{1}{n}}}\) , and \(\sqrt[n]{x }×\sqrt[n]{y }=\sqrt[n]{xy }\)

## Examples

### Simplifying Radical Expressions – Example 1:

Find the square root of \(\sqrt{144x^2 }\).

**Solution**:

Find the factor of the expression \(144x^2: 144=12×12\) and \(x^2=x×x\), now use radical rule: \(\sqrt[n]{a^n}=a\), Then: \(\sqrt{12^2 }=12\) and \(\sqrt{x^2} =x\) Finally: \(\sqrt{144x^2}=\sqrt{12^2}×\sqrt{x^2}=12×x=12x\)

### Simplifying Radical Expressions – Example 2:

Write this radical in exponential form. \(\sqrt[3]{x^4}\)

**Solution**:

To write a radical in exponential form, use this rule: \(\sqrt[n]{x^a}=x^{\frac{a}{n}}\) Then: \(\sqrt[3]{x^4}=x^{\frac{4}{3}}\)

### Simplifying Radical Expressions – Example 3:

Simplify. \(\sqrt{8x^3 }\)

**Solution**:

First factor the expression \(8x^3: 8x^3=2^3×x×x ×x\), we need to find perfect squares: \(8x^3=2^2×2×x^2×x=2^2×x^2×2x\), Then: \(\sqrt{8x^3 } =\sqrt{2^2 ×x^2}×\sqrt{2x}\)

Now use radical rule: \(\sqrt[n]{a^n}=a\), Then: \(\sqrt{2^2 ×x^2 }×\sqrt{(2x)}=2x×\sqrt{2x}=2x\sqrt{2x}\)

## Exercises for Simplifying Radical Expressions

- \(\color{blue}{\sqrt{121x^4}}\)
- \(\color{blue}{\sqrt{225x^6}}\)
- \(\color{blue}{\sqrt{36x^2}×\sqrt{64}}\)
- \(\color{blue}{\sqrt{4x^2}×\sqrt{81y^2}}\)
- \(\color{blue}{\sqrt{25x}×\sqrt{100x}}\)
- \(\color{blue}{\sqrt{49x^3}}\)

- \(\color{blue}{11x^2}\)
- \(\color{blue}{15x^3}\)
- \(\color{blue}{48x}\)
- \(\color{blue}{18xy}\)
- \(\color{blue}{50x}\)
- \(\color{blue}{7x\sqrt{x}}\)