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.

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?

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!.

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

Course: Algebra 1 (FL B.E.S.T.) > Unit 7

Lesson 6: Rewriting expressions with radicals

Simplifying square roots review

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

\sqrt{75}

by removing all perfect squares from inside the square root.

We start by factoring

75

, looking for a perfect square:

75 = 5 \times 5 \times 3 = 5^{2} \times 3

We found one! This allows us to simplify the radical:

\begin{aligned} \sqrt{75} & = \sqrt{5^{2} \cdot 3} \\ = \sqrt{5^{2}} \cdot \sqrt{3} \\ = 5 \cdot \sqrt{3} \end{aligned}

\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{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} = {(x^{3})}^{2} \cdot x

And now we can simplify:

\begin{aligned} \sqrt{54 x^{7}} & = \sqrt{3^{2} \cdot 6 \cdot {(x^{3})}^{2} \cdot x} \\ = \sqrt{3^{2}} \cdot \sqrt{6} \cdot \sqrt{{(x^{3})}^{2}} \cdot \sqrt{x} \\ = 3 \cdot \sqrt{6} \cdot x^{3} \cdot \sqrt{x} \\ = 3 x^{3} \sqrt{6 x} \end{aligned}

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

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

Want to join the conversation?

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skilavera1
Posted 5 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...”
(16 votes)
Answer
- Beaniebopbunyip
  Posted 5 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...”
  (65 votes)
Jaidyn McPherson
Posted 6 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
(14 votes)
Answer
- AD Baker
  Posted 5 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 ...”
  (51 votes)
dylan.forr99
Posted a year ago. Direct link to dylan.forr99's post “Anyone else need to take ...”
Anyone else need to take like 4 or 5 hours to really get a firm understanding of this lesson? or am I just dumb?
Button navigates to signup pageComment on dylan.forr99's post “Anyone else need to take ...”
(17 votes)
Answer
- Gina
  Posted a year ago. Direct link to Gina's post “Sometimes things snap rig...”
  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.
  Button navigates to signup page
  (5 votes)
Ha
Posted 6 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
(10 votes)
Answer
- Redapple8787
  Posted 6 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
  (14 votes)
bjacobsen4427
Posted a year ago. Direct link to bjacobsen4427's post “golly gracious i think iv...”
golly gracious i think ive passed out 15 times trying to these
Button navigates to signup pageComment on bjacobsen4427's post “golly gracious i think iv...”
(19 votes)
Answer
Lateo
Posted 7 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...”
(11 votes)
Answer
will.lindner.student
Posted 6 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
(5 votes)
Answer
- David Lee
  Posted 6 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
  (9 votes)
Misheel
Posted 3 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 ...”
(3 votes)
Answer
- Kim Seidel
  Posted 3 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...”
  (12 votes)
daniel wauchope
Posted 4 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 4 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
  (17 votes)
kjohnson8937
Posted 3 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
(5 votes)
Answer
- David Severin
  Posted 3 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
  (6 votes)