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College Algebra

Course: College Algebra>Unit 7

Lesson 2: 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.

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?
• 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?
• How can we solve a radical equation with another radical inside of it?
• So are you saying something like √ (√x) = 2? This is a very simple one, so square both sides to get √x=4, do it a second time to get x = 16. The alternate way is to go into rational exponents so if you have the cube root of the square root of (x-5) =2, you get ((x-5)^(1/2))^1/3 = 2, power to power requires multiplication, so (x-5)^1/6 = 2, opposite of 1/6 is 6 in exponent, so (x-5)^(1/6*6)=2^6, x-5=64, x=69. It obviously can get much more complicated than this.
• what about problems with a number already multiplying the square root. Do you multiply or add the numbers together?
• It's easier to understand if there is an example. Let's say you have √98. 98 is 49*2 which is 7^2*2, it would be 7√2. If you have 2√98. √98 is 7√2, 2√98 would be 14√2.

Hope this helps! If you have any questions or need help, please ask! :)
• how would we solve x!=120
• Factorials are based on multiplying all numbers below the number, so start dividing out starting at 2 until you reach the number you want. So 120/2=60/3=20/4=5. Answer is 5!.
• how do you do long division
• let me show you an example: 129/3
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