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: 8th grade (Illustrative Mathematics)>Unit 8

Lesson 1: Lesson 2: Side lengths and areas

# Square roots of perfect squares

Learn how to find the square root of perfect squares like 25, 36, and 81.
Let's start by taking a look at an example evaluating the square root of start color #1fab54, 25, end color #1fab54:
square root of, start color #1fab54, 25, end color #1fab54, end square root, equals, question mark
Step 1: Ask, "What number squared equals start color #1fab54, 25, end color #1fab54?"
Step 2: Notice that start color #11accd, 5, end color #11accd squared equals start color #1fab54, 25, end color #1fab54.
start color #11accd, 5, end color #11accd, squared, equals, start color #11accd, 5, end color #11accd, times, start color #11accd, 5, end color #11accd, equals, start color #1fab54, 25, end color #1fab54
square root of, start color #1fab54, 25, end color #1fab54, end square root, equals, start color #11accd, 5, end color #11accd
Here's a question to make sure you understood:
How can we be sure that start color #11accd, 5, end color #11accd is the right answer?

# Connection to a square

Finding the square root of start color #1fab54, 25, end color #1fab54 is the same as finding the side length of a square with an area of start color #1fab54, 25, end color #1fab54.
A square with an area of start color #1fab54, 25, end color #1fab54 has a side length of start color #11accd, 5, end color #11accd.

# Practice Set 1:

Problem 1A
• Current
4, squared, equals

### Reflection question

Which claim shows how square roots work?

# Practice Set 2:

Problem 2A
• Current
square root of, 1, end square root, equals

# Practice Set 3:

Problem 3A
• Current
square root of, 121, end square root, equals

## Want to join the conversation?

• Can an exponent be a negative number?
• You already know that an exponent represents the number of times you have to multiply a number by itself. A negative exponent is equivalent to the inverse of the same number with a positive exponent. There is nothing special about solving a problem that includes negative exponentials.
• That is awesome for you continue to learn ahead
• What happens if try to find the
the square root of an imperfect square
• Most probably it will be an irrational number in which case we can only approximate its value.

However, sometimes we can write the radicand as a fraction of two perfect squares.
Example: √5.76 = √(576∕100) = √(144∕25) = √(12²∕5²) = 12∕5 = 2.4
• I just noticed something really interesting (I think):

If I can't remember a square of some numbers (7^2 and 8^2 can be a bit tricky for me for some reason) but I remember the square number of the root that comes before it (6 and 36 in the case I'm trying to find 7^2),
I can do 36+6 to make it into 6*7, and then add a 7 to make it into a 7*7.

I tried to play around and find a rule and I think I found the formula:
n^2 + 2n + 1 = ( n + 1 )^2
(where n is the root number of the square that you do know).

if you would visualize the numbers on a grid, the n^2 is the area of the square, while the 2n+1 is the number of the additional units the is added on the side.

It's easier to see it on a times table: When looking at 25, the number diagonally next to it is 36. If you count the 'units' (the other multiplies that are on the same axes that leads toward 36 IE: 3,6,9 up to 36 on both sides) they will be the same as the 2n+1.

I tried to find a formula for a square of a root that isn't immediately follows the root I know. example: 3^2 = 9, 5^2 = ?.
unfortunately I couldn't think on one consistent formula, because there is always a need to add more and (n+1) with more additional 1s the further the number is.

does the formula I found have a name? I'm pretty sure I wasn't the first to think of that lol
• You can use the pascal's triangle I suppose

Edit1: here: https://en.wikipedia.org/wiki/Polynomial_expansion

Edit2: Sorry that is not true. That applies to expansions with higher powers, but not between different sqaures.
But maybe you can look at sum of squares where
∑x^2 = n(n+1)(2n+1)/6
• So, a perfect square is basically the answer to an exponent? (ex. 2^2(4), 12^12(144))
• it is an integer with a square root that is also an integer
• How can we use square roots in life?
• There are many ways in life you can use square roots. Some of them include a mathmatician, engineer, architect, carpenter, etc.
• [Hook]
'Cause every little thing is gonna be alright"
Singing, "Don't worry about a thing
'Cause every little thing is gonna be alright!"

[Chorus]
Rise up this morning, smiled with the rising sun
Three little birds pitch by my doorstep
Singing sweet songs of melodies pure and true
Saying, "This is my message to you-ou-ou"

[Hook]
Singing, "Don't worry about a thing
'Cause every little thing is gonna be alright"
Singing, "Don't worry about a thing (Don't worry)
'Cause every little thing is gonna be alright!"

[Chorus]
Rise up this morning, smiled with the rising sun
Three little birds pitch by my doorstep
Singing sweet songs of melodies pure and true
Saying, "This is my message to you-ou-ou"

[Hook]
'Cause every little thing is gonna be alright (Don't worry)
Singing, "Don't worry about a thing (I won't worry)
'Cause every little thing is gonna be alright!"
Singing, "Don't worry about a thing
'Cause every little thing is gonna be alright (I won't worry)
Singing, "Don't worry about a thing
'Cause every little thing is gonna be alright!"
'Cause every little thing is gonna be alright