# Simplifying square roots review

CCSS Math: HSN.RN.A.2
Learn how to rewrite square roots (and expressions containing them) so there's no perfect square within the square root. For example, rewrite √75 as 5⋅√3.

## Simplifying square roots

### Example

Let's simplify $\sqrt{75}$ by removing all perfect squares from inside the square root.
We start by factoring $75$, looking for a perfect square:
$75=5\times5\times3=\blueD{5^2}\times3$.
We found one! This allows us to simplify the radical:
\begin{aligned} \sqrt{75}&=\sqrt{\blueD{5^2}\cdot3} \\\\ &=\sqrt{\blueD{5^2}} \cdot \sqrt{{3}} \\\\ &=5\cdot \sqrt{3} \end{aligned}
So $\sqrt{75}=5\sqrt{3}$.
Want another example like this? Check out this video.

### Practice

Problem 1.1
Simplify.
Remove all perfect squares from inside the square root.
${\sqrt[]{12}}=$
Want to try more problems like these? Check out this exercise.

## Simplifying square roots with variables

### Example

Let's simplify $\sqrt{54x^7}$ by removing all perfect squares from inside the square root.
First, we factor $54$:
$54=3\cdot 3\cdot 3\cdot 2=3^2\cdot 6$
Then, we find the greatest perfect square in $x^7$:
$x^7=\left(x^3\right)^2\cdot x$
And now we can simplify:
\begin{aligned} \sqrt{54x^7}&=\sqrt{3^2\cdot 6\cdot\left(x^3\right)^2\cdot x} \\\\ &=\sqrt{3^2}\cdot \sqrt6 \cdot\sqrt{\left(x^3\right)^2}\cdot \sqrt x \\\\ &=3\cdot\sqrt6\cdot x^3\cdot\sqrt x \\\\ &=3x^3\sqrt{6x} \end{aligned}

### Practice

Problem 2.1
Simplify.
Remove all perfect squares from inside the square root.
$\sqrt{20x^8}=$
Want to try more problems like these? Check out this exercise.

## More challenging square root expressions

Problem 3.1
Simplify.
Combine like terms and remove all perfect squares from inside the square roots.
$2\sqrt{7x}\cdot 3\sqrt{14x^2}=$
Want to try more problems like these? Check out this exercise.