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.

How do I look at this and not cry?

This is the review. Why not start back at the beginning of the lesson and make sure you understand each part as you go. Then, retry the review.

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?

A fraction can be an exponent. When a fraction is an exponent, you can change it so that a there is a first, second, third, etc. root of something. For example, 1^1/2 = square root of 1 1^1/3 = third root of 1 1^1/4 = fourth root of 1 And so on and so forth. This was covered in a series of videos in the topic Rational Exponents and Radicals. https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-rational-exp-eval/v/fractional-exponents-with-numerators-other-than-1

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 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.

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.

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 ```

Main content

Algebra 1

Course: Algebra 1 > Unit 11

Lesson 3: Simplifying square roots

Simplifying square roots review

CCSS.Math:

HSN.RN.A.2

Google Classroom

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?

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skilavera1
Posted 4 years ago. Direct link to skilavera1's post “what grade maths would th...”
what grade maths would this be?
Button navigates to signup pageComment on skilavera1's post “what grade maths would th...”
(12 votes)
Answer
- Beaniebopbunyip
  Posted 4 years ago. Direct link to Beaniebopbunyip's post “I think it’s about eighth...”
  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.
  Comment on Beaniebopbunyip's post “I think it’s about eighth...”
  (51 votes)
Jaidyn McPherson
Posted 5 years ago. Direct link to Jaidyn McPherson's post “when will we ever use thi...”
when will we ever use this in everyday life? whats the point of even learning this?
Button navigates to signup pageButton navigates to signup page
(8 votes)
Answer
- AD Baker
  Posted 4 years ago. Direct link to AD Baker's post “Jaidyn, After learning ...”
  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.)
  Comment on AD Baker's post “Jaidyn, After learning ...”
  (41 votes)
Ha
Posted 5 years ago. Direct link to Ha's post “can a fraction be an expo...”
can a fraction be an exponent?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- Redapple8787
  Posted 5 years ago. Direct link to Redapple8787's post “A fraction can be an expo...”
  A fraction can be an exponent. When a fraction is an exponent, you can change it so that a there is a first, second, third, etc. root of something.
  For example,
  1^1/2 = square root of 1
  1^1/3 = third root of 1
  1^1/4 = fourth root of 1
  And so on and so forth. This was covered in a series of videos in the topic Rational Exponents and Radicals.
  https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-rational-exp-eval/v/fractional-exponents-with-numerators-other-than-1
  Button navigates to signup page
  (9 votes)
AlexC
Posted a month ago. Direct link to AlexC's post “How do I look at this and...”
How do I look at this and not cry?
Button navigates to signup pageComment on AlexC's post “How do I look at this and...”
(9 votes)
Answer
- Kim Seidel
  Posted a month ago. Direct link to Kim Seidel's post “This is the review. Why ...”
  This is the review. Why not start back at the beginning of the lesson and make sure you understand each part as you go. Then, retry the review.
  Button navigates to signup page
  (5 votes)
Lateo
Posted 6 years ago. Direct link to Lateo's post “In the video "Simplifying...”
In the video "Simplifying square roots (variables)" @
1:40
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?
Button navigates to signup pageComment on Lateo's post “In the video "Simplifying...”
(9 votes)
Answer
will.lindner.student
Posted 5 years ago. Direct link to will.lindner.student's post “what about problems with ...”
what about problems with a number already multiplying the square root. Do you multiply or add the numbers together?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- David Lee
  Posted 5 years ago. Direct link to David Lee's post “It's easier to understand...”
  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! :)
  Button navigates to signup page
  (7 votes)
daniel wauchope
Posted 3 years ago. Direct link to daniel wauchope's post “how do you do long divisi...”
how do you do long division
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- jonahfish
  Posted 3 years ago. Direct link to jonahfish's post “let me show you an exampl...”
  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
  Button navigates to signup page
  (16 votes)
Misheel
Posted 2 years ago. Direct link to Misheel's post “Can i also simplify √72 ...”
Can i also simplify √72 in this way: √72 = √9*8 = √9*√8 = 3√8

instead of: √72 = √2*36 = √36*√2 = 6√2
Button navigates to signup pageComment on Misheel's post “Can i also simplify √72 ...”
(2 votes)
Answer
- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “Yes, you can take that ap...”
  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.
  Comment on Kim Seidel's post “Yes, you can take that ap...”
  (9 votes)
yanchu m
Posted 3 years ago. Direct link to yanchu m's post “How can we solve a radica...”
How can we solve a radical equation with another radical inside of it?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- David Severin
  Posted 3 years ago. Direct link to David Severin's post “So are you saying somethi...”
  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.
  Comment on David Severin's post “So are you saying somethi...”
  (8 votes)
kjohnson8937
Posted 2 years ago. Direct link to kjohnson8937's post “how would we solve x!=120”
how would we solve x!=120
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- David Severin
  Posted 2 years ago. Direct link to David Severin's post “Factorials are based on m...”
  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!.
  Button navigates to signup page
  (4 votes)