 ## 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: $$\color{blue}{\sqrt{x}+\sqrt{x}=2\sqrt{x} }$$, $$\color{blue}{2\sqrt{x}-\sqrt{x}=\sqrt{x} }$$

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

### 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: $$\sqrt{3^2 }×\sqrt{5^2 }=3×5=15$$

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

### 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

### 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. $$\color{blue}{3\sqrt{10}}$$ 