In this article, you learn how to simplify radicals and how to do mathematics operations with radicals.

## Related Topics

- How to Solve Powers of Products and Quotients
- How to Multiply Exponents
- How to Divide Exponents
- How to Solve Zero and Negative Exponents
- How to Solve Negative Exponents and Negative Bases

## Step by step guide to solve radicals

- A square root (radical) of \(x\) is a number \(r\) whose square is: \(r^2=x\)

\(r\) is a square root of \(x\). - A cube root of \(x\) is a number \(r\) whose cube is: \(r^3=x\)

\(r\) is a cube root of \(x\). - Radical rules: \(\color{blue}{\sqrt[n]{a^n }=a}\), \(\color{blue}{ \sqrt{x} \times \sqrt{y}= \sqrt{xy} } \)
- We can add or subtract radicals when they have exactly the same value under radicals: \(\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} } \), \(\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} } \)

### Radicals – Example 1:

Find the square root of \(\sqrt{169}\).

**Solution:**

First factor the number: \(169=13^2\), Then: \(\sqrt{169}=\sqrt{13^2 }\)

Now use radical rule: \(\color{blue}{\sqrt[n]{a^n }=a}\)

Then: \(\sqrt{13^2 }=13\)

### Radicals – Example 2:

Evaluate. \(\sqrt{9} \times \sqrt{25}=\)

**Solution:**

First factor the numbers: \(9=3^2\) and \(25=5^2\)

Then: \(\sqrt{9}×\sqrt{25}=\sqrt{3^2 }×\sqrt{5^2 }\)

Now use radical rule: \(\color{blue}{\sqrt[n]{a^n }=a}\) , Then: \(\sqrt{3^2 }×\sqrt{5^2 }=3×5=15\)

### Radicals – Example 3:

Find the square root of \(\sqrt{225}\).

**Solution:**

First factor the number: \(225=15^2\), Then: \(\sqrt{225}=\sqrt{15^2}\)

Now use radical rule: \(\color{blue}{\sqrt[n]{a^n }=a}\) ,Then: \(\sqrt{15^2}=15\)

### Radicals – Example 4:

Evaluate. \(2\sqrt{3}-\sqrt{48}= \)

**Solution:**

There are different values under radical signs. Let’s simplify \(\sqrt{48}\). \(48\) can be written as \(16×3\). We can write \(\sqrt{48}\) as \(\sqrt{ 16×3}\) or \(\sqrt{ 16}×\sqrt{3}\). \(\sqrt{ 16}=4\), then:

\(\sqrt{ 48}=4\sqrt{3}\). Now, we can solve \(2\sqrt{3}-\sqrt{48}=2\sqrt{3}-4\sqrt{3}= \ -2\sqrt{3}\)

## Exercises for Solving Radicals

### Find the value each square root.

- \(\color{blue}{\sqrt{1}}\)
- \(\color{blue}{ \sqrt{4} }\)
- \(\color{blue}{ \sqrt{9} }\)
- \(\color{blue}{ \sqrt{900} }\)
- \(\color{blue}{ \sqrt{529} }\)
- \(\color{blue}{ \sqrt{90} }\)

### Download Square Roots Worksheet

- \(\color{blue}{1}\)
- \(\color{blue}{2}\)
- \(\color{blue}{3}\)
- \(\color{blue}{30}\)
- \(\color{blue}{23}\)
- \(\color{blue}{3\sqrt{10}}\)