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## Algebra 1

### Course: Algebra 1>Unit 11

Lesson 3: Simplifying square roots

# Simplifying square roots review

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 square root of, 75, end square root by removing all perfect squares from inside the square root.
We start by factoring 75, looking for a perfect square:
75, equals, 5, times, 5, times, 3, equals, start color #11accd, 5, squared, end color #11accd, times, 3.
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 square root of, 75, end square root, equals, 5, square root of, 3, end square root.
Want another example like this? Check out this video.

### Practice

Problem 1.1
• Current
Simplify.
Remove all perfect squares from inside the square root.
root, start index, end index, equals

Want to try more problems like these? Check out this exercise.

## Simplifying square roots with variables

### Example

Let's simplify square root of, 54, x, start superscript, 7, end superscript, end square root by removing all perfect squares from inside the square root.
First, we factor 54:
54, equals, 3, dot, 3, dot, 3, dot, 2, equals, 3, squared, dot, 6
Then, we find the greatest perfect square in x, start superscript, 7, end superscript:
x, start superscript, 7, end superscript, equals, left parenthesis, x, cubed, right parenthesis, squared, dot, 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
• Current
Simplify.
Remove all perfect squares from inside the square root.
square root of, 20, x, start superscript, 8, end superscript, end square root, equals

Want to try more problems like these? Check out this exercise.

## More challenging square root expressions

Problem 3.1
• Current
Simplify.
Combine like terms and remove all perfect squares from inside the square roots.
2, square root of, 7, x, end square root, dot, 3, square root of, 14, x, squared, end square root, equals

Want to try more problems like these? Check out this exercise.