How to Simplify Radical Expressions? (+FREE Worksheet!)

Radical Expressions in many equations must be simplified in a special way, and 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: $\sqrt{144x^2}=\sqrt{12^2}×\sqrt{x^2}=12×x=12x$

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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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