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## How to Solve Square Root Problems

Last Updated: March 7, 2023 References

## Understanding Squares and Square Roots

- As another example, let's find the square root of 25 (√(25)). This means we want to find the number that squares to make 25. Since 5 2 = 5 × 5 = 25, we can say that √(25) = 5 .
- You can also think of this as "undoing" a square. For example, if we want to find √(64), the square root of 64, let's start by thinking of 64 as 8 2 . Since a square root symbol basically "cancels out" a square, we can say that √(64) = √(8 2 ) = 8 .

- 1 2 = 1 × 1 = 1
- 2 2 = 2 × 2 = 4
- 3 2 = 3 × 3 = 9
- 4 2 = 4 × 4 = 16
- 5 2 = 5 × 5 = 25
- 6 2 = 6 × 6 = 36
- 7 2 = 7 × 7 = 49
- 8 2 = 8 × 8 = 64
- 9 2 = 9 × 9 = 81
- 10 2 = 10 × 10 = 100
- 11 2 = 11 × 11 = 121
- 12 2 = 12 × 12 = 144

- Let's say that we want to find the square root of 900. At first glance, this looks very difficult! However, it's not hard if we separate 900 into its factors. Factors are the numbers that can multiply together to make another number. For instance, since you can make 6 by multiplying 1 × 6 and 2 × 3, the factors of 6 are 1, 2, 3, and 6.
- Instead of working with the number 900, which is somewhat awkward, let's instead write 900 as 9 × 100. Now, since 9, which is a perfect square, is separated from 100, we can take its square root on its own. √(9 × 100) = √(9) × √(100) = 3 × √(100). In other words, √(900) = 3√(100) .
- We can even simplify this two steps further by dividing 100 into the factors 25 and 4. √(100) = √(25 × 4) = √(25) × √(4) = 5 × 2 = 10. So, we can say that √(900) = 3(10) = 30 .

## Using Long Division-Style Algorithms

- Start by writing out your square root problem in the same from as a long division problem. For example, let's say that we want to find the square root of 6.45, which is definitely not a convenient perfect square. First, we'd write an ordinary radical symbol (√), then we'd write our number underneath it. Next, we'd make a line above our number so that it's in a little "box" — just like in long division. When we're done, we should have a long-tailed "√" symbol with 6.45 written under it.
- We'll be writing numbers above our problem, so be sure to leave space.

## Quickly Estimating Imperfect Squares

- Multiply 6.4 by itself to get 6.4 × 6.4 = 40.96 , which is slightly higher than original number.
- Next, since we over-shot our answer, we'll multiply the number one tenth less than our estimate above by itself and to get 6.3 × 6.3 = 39.69 . This is slightly lower than our original number. This means that the square root of 40 is somewhere between 6.3 and 6.4 . Additionally, since 39.69 is closer to 40 than 40.96, you know the square root will be closer to 6.3 than 6.4.

## Calculator, Practice Problems, and Answers

## Expert Q&A Did you know you can get expert answers for this article? Unlock expert answers by supporting wikiHow

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## You Might Also Like

- ↑ David Jia. Academic Tutor. Expert Interview. 14 January 2021.
- ↑ https://virtualnerd.com/algebra-foundations/powers-square-roots/powers-exponents/squaring-a-number
- ↑ https://www.mathsisfun.com/square-root.html
- ↑ http://virtualnerd.com/algebra-1/algebra-foundations/powers-square-roots/square-roots/square-root-estimation
- ↑ https://www.cuemath.com/algebra/perfect-squares/
- ↑ https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-simplify-square-roots/a/simplifying-square-roots-review
- ↑ https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/00%3A_Preliminary_Topics_for_College_Algebra/0.03%3A_Review_-_Radicals_(Square_Roots)
- ↑ http://www.homeschoolmath.net/teaching/square-root-algorithm.php
- ↑ https://www.cuemath.com/algebra/square-root-by-long-division-method/
- ↑ https://virtualnerd.com/algebra-1/algebra-foundations/powers-square-roots/square-roots/square-root-estimation
- ↑ http://www.math.com/students/calculators/source/square-root.htm

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## Square Root

## What is Square Root?

## Square Root Definition

## How to Find Square Root?

## Repeated Subtraction Method of Square Root

You can observe that we have subtracted 4 times. Thus,√16 = 4

- Step 1: Divide the given number into its prime factors .
- Step 2: Form pairs of similar factors such that both factors in each pair are equal.
- Step 3: Take one factor from the pair.
- Step 4: Find the product of the factors obtained by taking one factor from each pair.
- Step 5: That product is the square root of the given number.

Let us find the square root of 144 by this method.

This method works when the given number is a perfect square number.

This is a very long process and time-consuming.

- Step 1: Place a bar over every pair of digits of the number starting from the units' place (right-most side). We will have two pairs, i.e., 1 and 80
- Step 2: We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

## Square Root Table

The square roots of numbers that are not perfect squares are irrational numbers .

## Square Root Formula

## How to Simplify Square Root?

For fractions, there is also a similar rule: √x/√y = √(x/y). For example: √50/√10 = √(50/10)= √5

## Square Root of a Negative Number

## Square of a Number

## How to Find the Square of a Number?

## Squares and Square Roots

- When "square" is removed from one side of the equation, we get the square root on the other side. For example, 4 2 = 16 means, 4 = √16. This is also known as "taking square root on both sides".
- When "square root" is removed from one side of the equation, we get square on the other side. For example, √25 = 5 means, 25 = 5 2 . This is also known as "squaring on both sides".

This logic helps in solving many equations in algebra. Consider the following example:

Example: Solve the equation √(2x + 3) = 10.

## Square Root of Numbers

Example 1: Find out the square root of 529 by the prime factorization method.

We can see that, 529 = 23×23 ⇒ √529= 23

Answer: a) 5 b) 256 c) 400 d) 20

Example 3: Determine the square root of 60.

From prime factorization of 60 , we get,

Therefore, Square root of 60 = 2√15

go to slide go to slide go to slide

## Practice Questions on Square Root

## FAQs on Square Root

- Square Root 1 to 10
- Square Root 1 to 20
- Square Root 1 to 25
- Square Root 1 to 30
- Square Root 1 to 50
- Square Root 1 to 100

## How to Find the Square Root of a Number?

- Repeated Subtraction Method
- Prime Factorization Method
- Estimation and Approximation Method
- Long Division Method

## How to Find the Square Root of a Decimal Number?

## Can Square Root be Negative?

## What is the Square Root Symbol?

## What is the Formula for Calculating the Square Root of a Number?

## What is the Square and Square Root of a Number?

## Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?

## How to Find a Square Root on a Calculator?

- Fraction Square Root Calculator
- Adding Square Roots Calculator
- Multiplying Square Roots Calculator
- Simplify Square Roots Calculator

## How to Multiply Two Square Root Values Together?

## What are the Applications of the Square Root Formula?

There are various applications of the square root formula:

- The square root formula is mainly used in algebra and geometry .
- It helps in finding the roots of a quadratic equation .
- It is widely used by engineers.

## What does the Square of a Number mean?

## How to Find the Square Root of a Negative Number?

## Why is the Square of a Negative Number Positive?

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## Unit 11: Lesson 3

- Simplifying square roots (variables)
- Simplifying square-root expressions
- Simplifying square roots review
- Exponents & radicals: FAQ

## Want to join the conversation?

## Video transcript

## Sciencing_Icons_Science SCIENCE

## The Basics of Square Roots (Examples & Answers)

## How to Solve a Square Root Equation

## TL;DR (Too Long; Didn't Read)

You can factor square roots just like ordinary numbers, so √ ab = √ a √ b , or √6 = √2√3.

However, 66 is also divisible by 2, so you can write:

In short, you can simplify square roots using the following rules

## The square root of 8

## The square root of 4

## The square root of 12

## The square root of 20

The square root of 20 can be found in the same way:

## The square root of 32

Finally, tackle the square root of 32 using the same approach:

Here, note that we already calculated the square root of 8 as 2√2, and that √4 = 2, so:

Try to solve these before looking at the answers below:

## Related Articles

## Find Your Next Great Science Fair Project! GO

We Have More Great Sciencing Articles!

## How to Use PEMDAS & Solve With Order of Operations (Examples)

## How to find the square root of a number and calculate it by hand

## STEP 1: Separate The Digits Into Pairs

In our case here, 2,025 becomes 20 25 .

## STEP 2: Find The Largest Integer

## STEP 3: Now Subtract That Integer

## STEP 4: Let's Move To The Next Pair

## STEP 5: Find The Right Match

Write 5 next to 4 in the top right corner. It is the second digit in the root.

## STEP 6: Subtract Again

The next example explains what I mean.

## EXAMPLE: We dig deeper...

This time the number consists of an odd number of digits including the ones after the decimal point.

If you read this far, tweet to the author to show them you care. Tweet a thanks

## Simplifying Square Roots

## Example: √12 is simpler as 2√3

Get your calculator and check if you want: they are both the same value!

Here is the rule: when a and b are not negative

## Example: simplify √12

And the square root of 4 is 2:

## Example: simplify √8

(Because the square root of 4 is 2)

## Example: simplify √18

√18 = √(9 × 2) = √9 × √2 = 3√2

It often helps to factor the numbers (into prime numbers is best):

## Example: simplify √6 × √15

First we can combine the two numbers:

Then we see two 3s, and decide to "pull them out":

There is a similar rule for fractions:

## Example: simplify √30 / √10

## Some Harder Examples

Example: simplify √20 × √5 √2.

See if you can follow the steps:

## Example: simplify 2√12 + 9√3

Now both terms have √3, we can add them:

Note: a root we can't simplify further is called a Surd . So √3 is a surd. But √4 = 2 is not a surd.

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## IMAGES

## VIDEO

## COMMENTS

According to Saint Louis University, the ancient Egyptians created the square root and most likely used it for architecture, building pyramids and other daily activities that required math. Most of the present-day knowledge of Egyptian math...

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This math video tutorial explains how to simplify square roots.My E-Book: https://amzn.to/3B9c08zVideo Playlists: https://www.video-tutor.

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First, we will find the square root of the numbers a and b. Then, after finding the square root we will multiply the square roots value together. Let us

So let me explain. Let's assume x is positive. x^2=x*x , of course. But -x*-x=x^2. You can try this with any number. Let's say x^2=y. So the square root of y

Roots are nice, but we prefer dealing with regular numbers as much as possible. So, for example, instead of √4 we prefer dealing with 2.

The “√” symbol tells you to take the square root of a number, and you can find this on most calculators.

STEP 1: Separate The Digits Into Pairs · STEP 2: Find The Largest Integer · STEP 3: Now Subtract That Integer · STEP 4: Let's Move To The Next Pair.

A Fun Way to Calculate a Square Root ; a) start with a guess (let's guess 4 is the square root of 10) ; around, b) divide by the guess (10/4 = 2.5) c) add that to

Simplifying Square Roots · Example: √12 is simpler as 2√3. Get your calculator and check if you want: they are both the same value! · Example: simplify √12. 12

If you want the answer to be a whole number, choose "perfect squares," which makes the radicand to be a perfect square (1, 4, 9, 16, 25, etc.). If you choose to