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## Pre-algebra

### Unit 11: Lesson 1

Square roots & cube roots- Intro to square roots
- Square roots of perfect squares
- Square roots
- Intro to cube roots
- Cube roots
- Worked example: Cube root of a negative number
- Equations with square roots & cube roots
- Square root of decimal
- Roots of decimals & fractions
- Equations with square roots: decimals & fractions
- Dimensions of a cube from its volume
- Square and cube challenge
- Square roots review
- Cube roots review

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# Square roots review

CCSS.Math:

Review square roots, and try some practice problems.

### Square roots

The square root of a number is the factor that we can multiply by itself to get that number.

The symbol for square root is square root of, end square root .

Finding the square root of a number is the opposite of squaring a number.

**Example:**

start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd or start color #11accd, 4, end color #11accd, squared equals, start color #1fab54, 16, end color #1fab54

So square root of, start color #1fab54, 16, end color #1fab54, end square root, equals, start color #11accd, 4, end color #11accd

If the square root is a whole number, it is called a perfect square! In this example, start color #1fab54, 16, end color #1fab54 is a perfect square because its square root is a whole number.

*Want to learn more about finding square roots? Check out this video.*

## Finding square roots

If we can't figure out what factor multiplied by itself will result in the given number, we can make a factor tree.

**Example:**

Here is the factor tree for 36:

So the prime factorization of 36 is 2, times, 2, times, 3, times, 3.

We're looking for square root of, 36, end square root, so we want to split the prime factors into two identical groups.

Notice that we can rearrange the factors like so:

So left parenthesis, 2, times, 3, right parenthesis, squared, equals, 6, squared, equals, 36.

So, square root of, 36, end square root is 6.

## Practice

*Want to try more problems like this? Check out this exercise:*Finding square roots

*Or this challenge exercise:*Equations with square and cube roots

## Want to join the conversation?

- When doing prime factorisation to get all perfect squares out of a square root, how do you decide which prime to factor out by?(18 votes)
- I don't know what you mean by "how do you decide which prime to factor out by??

1) Are you asking how do you start the prime factorization? You can start anywhere. you need any 2 numbers that multiply to the original number, and then keep factoring until you get the prime factors

2) Or, are you asking how do you know which prime factors are perfect squares? Any prime factor that occurs twice (is squared), is a perfect square.

If this doesn't answer your question, comment back and please clarify your question.(12 votes)

- I use Kumon as well as Khan Academy, and they are attempting to teach me a method I don't understand and can't find anywhere else. This is the procedure:

to find the square root of 3969:

1. Divide the radicand into groups of two digits, starting from the right side.

2. Find the number that is closest to but less that 39 when squared

3. Write the number found in step (2) twice and find the sum.

4.find the number, x, that is closest to but less than or equal to 369 when sustituted into 12x*x.

5. Write the number found in step (4)

Have you heard of this method before and if so can you explain it, please?(5 votes)- Idk what you thinking but Kumon does not teach anyone(6 votes)

- can negative integers have square and how(5 votes)
- Yes, negative integers can have square roots. The topic is imaginary numbers. For example, i^2=-1. If you want to learn more about this, you can search up imaginary numbers on Khan Academy and watch videos on them.(8 votes)

- How do you solve an equation with a square root in it(2 votes)
- Good question!

Generally, a square root equation is solved by isolating the square root (or radical), squaring both sides to get rid of the square root, and then solving the resulting equation. Solutions**must**be checked by substituting them into the**original**equation, because squaring both sides can create extraneous (invalid) solutions.

Example: Solve sqrt(x) + x = 0.

sqrt(x) = -x

x = x^2

0 = x^2 - x

0 = x(x - 1)

x = 0 or x - 1 = 0

x = 0 or x = 1

Check: sqrt(0) + 0 = 0 + 0 = 0, so x = 0 is a valid solution.

sqrt(1) + 1 = 1 + 1 = 2, which is not 0, so x = 1 is an extraneous (invalid) solution.

So the only solution is x = 0.

Have a blessed, wonderful day!(12 votes)

- Hey guys,106jmb i saw your question. I saw a faster way to find cube roots.

We already know some basic cube numbers

0^{3}0

3

=0

1^{3}1

3

=1

2^{3}2

3

=8

3^{3}3

3

=27

4^{3}4

3

=64

5^{3}5

3

=125

6^{3}6

3

=216

7^{3}7

3

=343

8^{3}8

3

=512

9^{3}9

3

=729

Now, the common thing here is that each ones digit of the cube numbers is the same number that is getting cubed , except for 2 ,8 ,3 ,7 .

now let us take a cube no like 226981 .

to see which is the cube root of that number , first check the last 3 digits that is 981 . Its last digit is 1 so therefore the last digit of the cube root of 226981 is 1 .

Now for the remaining digits that is 226

Now 226 is the nearer & bigger number compared to the cube of 6 (216)

So the cube root of 226981 is 61

Let us take another example - 148877

Here 7 is in the last digit but the cube of seven's last digit is not seven. But the cube of three has the last digit as 7.

So the last digit of the cube root of 148877 is 3.

Now for the remaining digits 148.

It is the nearer and bigger than the cube of 5 (125).

Therefore the cube root of 148877 is 53.

Let us take another example 54872.

Here the last three digit's (872) last digit is 2 but the cube of 2's last digit is not 2 but the last of the cube of 8 is 2.

So the last digit of the cube root of 54872 is 8.

Now of the remaining numbers (54). It is nearer and bigger to the cube of 3 (27). So therefore the cube root of 54872 is 38.(7 votes)- i also use this method!(1 vote)

- I need help this is very challanging!(3 votes)
- Is there anything specific you don't understand?(3 votes)

- 4 square root 3 squared(3 votes)
- That could be interpreted in two ways

4 (srt(3))^2 = 4*3=12

(4 sqrt(3))^2=16*3=48.(3 votes)

- Easy you need harder math(3 votes)
- Why is it both 10 and -10(1 vote)
- Remember that a negative times a negative equals a positive? 10 * 10 = 100, and -10 * -10 = 100.(3 votes)

- Do you know cube roots(1 vote)
- ★
**Finding the cube root is similar to finding the square root**,*but instead of grouping primes in twos,*, because instead we're looking for what the third power root is.**group matching primes by three**

No matter which prime you start with, the number will break down to the same primes.**Cubed root of 64**?**64**

/\**2**• 32

/\

**2**• 16

/\

**2**• 8

/\

**2**• 4

/\

**2**•**2**

•There are six 2s.

2 • 2 • 2 • 2 • 2 • 2 = 64

Which**group by three**.*twice*

(2 • 2 • 2) • (2 • 2 • 2)

•**Take one representative**, and multiply.*from each matching prime group*

2 • 2 =**4**←the root! 🥳

this tells us…

★.**The cubed root of 64 is 4**

because four cubed, 4^3 = 64

★**All the roots work this way**…

• find the prime factors

• group*matching primes*by the indexed number

• take one representative from each group and multiply them

So if we need to find the 5th root,*the matching primes would be grouped five each*.

(≧▽≦) I hope this helps!(3 votes)