If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### 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 $\sqrt{75}$ by removing all perfect squares from inside the square root.
We start by factoring $75$, looking for a perfect square:
$75=5×5×3={5}^{2}×3$.
We found one! This allows us to simplify the radical:
$\begin{array}{rl}\sqrt{75}& =\sqrt{{5}^{2}\cdot 3}\\ \\ & =\sqrt{{5}^{2}}\cdot \sqrt{3}\\ \\ & =5\cdot \sqrt{3}\end{array}$
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.
$\sqrt{\phantom{A}12}=$

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

## Simplifying square roots with variables

### Example

Let's simplify $\sqrt{54{x}^{7}}$ 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:
$\begin{array}{rl}\sqrt{54{x}^{7}}& =\sqrt{{3}^{2}\cdot 6\cdot {\left({x}^{3}\right)}^{2}\cdot x}\\ \\ & =\sqrt{{3}^{2}}\cdot \sqrt{6}\cdot \sqrt{{\left({x}^{3}\right)}^{2}}\cdot \sqrt{x}\\ \\ & =3\cdot \sqrt{6}\cdot {x}^{3}\cdot \sqrt{x}\\ \\ & =3{x}^{3}\sqrt{6x}\end{array}$

### Practice

Problem 2.1
Simplify.
Remove all perfect squares from inside the square root.
$\sqrt{20{x}^{8}}=$

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.
$2\sqrt{7x}\cdot 3\sqrt{14{x}^{2}}=$

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

## Want to join the conversation?

• what grade maths would this be?
• I think it’s about eighth or ninth grade. But people take math at different times. I’ve known fith graders who have taken algebra and geometry in the same year, and I’ve known ninth graders who have taken algebra. Even if you’re taking algebra in ninth grade, that’s okay. What really matters is that you understand the content when you learn it.
• when will we ever use this in everyday life? whats the point of even learning this?
• Jaidyn,

After learning this helps you pass your Math class and graduate high school, there are many careers where this is used. Most obviously, it's used in engineering and computer science. However, when I worked in construction, I used to use square roots regularly to determine whether items would fit through a doorway on a diagonal. (Note: this also involves trigonometry.)
• can a fraction be an exponent?
• Anyone else need to take like 4 or 5 hours to really get a firm understanding of this lesson? or am I just dumb?
• Sometimes things snap right into place and the light goes on right away, and other times we need review and practice. If you got this far, you already have all the pieces you need to work with radicals. It's a matter of seeing how they go together.
• golly gracious i think ive passed out 15 times trying to these
• In the video "Simplifying square roots (variables)" @ Sal explains "as I said in the last video, the principal root of X squared is going to be the absolute value of X, just in case X is a negative number". I have two questions:
(1) Can anybody please point me to that video? I can't find it.
(2) I don't understand the need for an absolute value. If we state, before beginning to solve the problem, that the domain of the X variable is the Positive Real Numbers (or X greater than or equal to zero), aren't we already cancelling out the possibility that the X variable assumes a negative value by restricting the domain, thus rendering the use of the absolute value unnecessary?
• Can i also simplify √72 in this way: √72 = √9*8 = √9*√8 = 3√8

instead of: √72 = √2*36 = √36*√2 = 6√2
• Yes, you can take that approach. But, your work is incomplete. When you simplify a square root, you need to ensure you have removed all perfect squares. With 3√8, you still have a perfect square inside the radical.
3√8 = 3√(4*2) = 3√4 * √2 = 3*2√2 = 6√2
Hope this helps.
• what about problems with a number already multiplying the square root. Do you multiply or add the numbers together?
   43    three "goes in" 43 times ______3|129      12      3 goes into 12 four times -12      minus 3*4   (in between 12 and 0, and 3,0 there is a line)   09    remainder of 0. bring down the 9    3