# How to Simplify Radical Expressions

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.

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

• Find the prime factors of the numbers or expressions inside the radical.
$$\sqrt[n]{x^a }=x^{\frac{a}{n}}, \sqrt[n]{xy }=x^{\frac{1}{n}}×y^{\frac{1}{n}}, \sqrt[n]{\frac{x}{y} }=\frac{x^{\frac{1}{n}}}{y^{\frac{1}{n}}}$$ , 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

1. $$\color{blue}{\sqrt{121x^4}}$$
2. $$\color{blue}{\sqrt{225x^6}}$$
3. $$\color{blue}{\sqrt{36x^2}×\sqrt{64}}$$
4. $$\color{blue}{\sqrt{4x^2}×\sqrt{81y^2}}$$
5. $$\color{blue}{\sqrt{25x}×\sqrt{100x}}$$
6. $$\color{blue}{\sqrt{49x^3}}$$
1. $$\color{blue}{11x^2}$$
2. $$\color{blue}{15x^3}$$
3. $$\color{blue}{48x}$$
4. $$\color{blue}{18xy}$$
5. $$\color{blue}{50x}$$
6. $$\color{blue}{7x\sqrt{x}}$$

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