# How to Solve Radicals? (+FREE Worksheet!) ## 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: $$\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} }$$, $$\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} }$$

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: $$\sqrt{169}=$$ $$\sqrt{13^2 }=13$$

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: $$\sqrt{9} \times \sqrt{25}=$$ $$\sqrt{3^2 }×\sqrt{5^2 }=3×5=15$$

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: $$\sqrt{225}=$$ $$\sqrt{15^2}=15$$

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 $$\sqrt{ 16}×\sqrt{3}$$. $$16= 4^2$$, Then: $$\sqrt{16}= \sqrt{4^2}=4$$
$$\sqrt{ 48}=4\sqrt{3}$$. Now, we can solve $$2\sqrt{3}-\sqrt{48}=2\sqrt{3}-4\sqrt{3}= \ -2\sqrt{3}$$

### Find the value each square root.

1. $$\color{blue}{\sqrt{1}}$$
2. $$\color{blue}{ \sqrt{4} }$$
3. $$\color{blue}{ \sqrt{9} }$$
4. $$\color{blue}{ \sqrt{900} }$$
5. $$\color{blue}{ \sqrt{529} }$$
6. $$\color{blue}{ \sqrt{90} }$$

1. $$\color{blue}{1}$$
2. $$\color{blue}{2}$$
3. $$\color{blue}{3}$$
4. $$\color{blue}{30}$$
5. $$\color{blue}{23}$$
6. $$\color{blue}{3\sqrt{10}}$$

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