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 75\sqrt{75} by removing all perfect squares from inside the square root.
We start by factoring 7575, looking for a perfect square:
75=5×5×3=52×375=5\times5\times3=\blueD{5^2}\times3.
We found one! This allows us to simplify the radical:
75=523=523=53\begin{aligned} \sqrt{75}&=\sqrt{\blueD{5^2}\cdot3} \\\\ &=\sqrt{\blueD{5^2}} \cdot \sqrt{{3}} \\\\ &=5\cdot \sqrt{3} \end{aligned}
So 75=53\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.
12={\sqrt[]{12}}=
Want to try more problems like these? Check out this exercise.

Simplifying square roots with variables

Example

Let's simplify 54x7\sqrt{54x^7} by removing all perfect squares from inside the square root.
First, we factor 5454:
54=3332=32654=3\cdot 3\cdot 3\cdot 2=3^2\cdot 6
Then, we find the greatest perfect square in x7x^7:
x7=(x3)2xx^7=\left(x^3\right)^2\cdot x
And now we can simplify:
54x7=326(x3)2x=326(x3)2x=36x3x=3x36x\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.
20x8=\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.
27x314x2=2\sqrt{7x}\cdot 3\sqrt{14x^2}=
Want to try more problems like these? Check out this exercise.
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