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Square and Square Roots Questions

Square and square root questions with solutions are given here for students to practice and memorise the square and square roots of different numbers. Often we have seen students face difficulty memorising the squares and square roots of numbers; one way to remember them could be by solving more and more questions so that they get well familiarised with the concepts of square and square roots.

Learn more about Squares and square roots .

Below is the list of the first 20 perfect squares and square roots:

Video Lesson on Square of a number

Square and Square Roots Questions with Solutions

Let us solve a few questions based on square and square roots.

Question 1:

Find the difference between the following without actual calculation:

(i) 35 2 – 34 2

(ii) 134 2 – 133 2

(iii) 62 2 – 61 2.

The difference between the squares of two consecutive natural numbers is the sum of the natural numbers.

(i) 35 2 – 34 2 = 35 + 34 = 69

(ii) 134 2 – 133 2 = 134 + 133 = 267

(iii) 62 2 – 61 2 = 62 + 61 = 123

Question 2:

Express the following squares as a sum of two consecutive natural numbers:

The square of any odd natural number m can be expressed as a sum of two consecutive natural numbers: (m 2 + 1)/2 and (m 2 – 1)/2.

(i) 17 2 = 289 = [(289 + 1)/2 + (289 – 1)/2] = 145 + 144

(ii) 23 2. = 529 = [(529 – 1)/2 + (529 + 1)/2] = 264 + 265

Question 3:

Evaluate the following:

(i) 89 × 91

(ii) 16 × 18

(i) 89 × 91 = (90 – 1)(90 + 1) = 90 2 – 1 = 8100 – 1

(ii) 16 × 18 = (17 – 1)(17 + 1) = 17 2 – 1

Question 4:

Find the square root of the following by prime factorisation:

(iii) 1.5625

(i) √7744 = √(2 × 2 × 2 × 2 × 2 × 2 × 11 × 11) = 2 × 2 × 2 × 11 = 88

(ii) √1156 = √(2 × 2 × 17 × 17) = 2 × 17 = 34

(iii) √1.5625 = √(15625/10000) = √(5 × 5 × 5 × 5 × 5 × 5)/√(2 × 2 × 2 × 2 × 5 × 5 × 5 × 5)

= (5 × 5 × 5)/ (100) = 1.25

Also, watch these videos to get a clear understanding of square roots and how to visualise them.

Question 5:

Find the square root of 21,25,764 by the long division method.

By the long division method:

∴ √2125764 = 1458

Also check: Trick to quickly find square root of any square number .

Question 6:

Find the value of x: √(x × 3042) + √(204.49) = 92.3.

First, we shall calculate √(204.49) = √(20449/100) = √(11 × 11 × 13 × 13)/√(100) = 14.3

∴ √(x × 3042) + √(204.49) = 92.3

⇒ √(x × 3042) + 14.3 = 92.3

⇒ √(x × 3042) = 92.3 – 14.3

⇒ √(x × 3042) = 78

Squaring both sides, we get:

3042x = 6084

Question 7:

Find the smallest number by which 10125 must be divided to get a perfect square number. Also, find the square root of the perfect square number.

Prime factorisation of 10125 = 5 × 5 × 5 × 9 × 9

We see that one 5 is left unpaired. Thus, 10125 must be divided by 5 to get the perfect square number.

10125 ÷ 5 = 2025

Now, √2025 = √(5× 5 × 9 × 9) = 5 × 9 = 45

Also check:

• Perfect Square Calculator
• Square Root Calculator
• Cube Root Questions for Practice

Question 8:

In an auditorium, there are 21025 chairs. If the number of rows is equal to the number of columns, find out how many rows are there.

Total number of chairs = 21025

Let x be the numbers of rows and columns of the chair, respectively.

Then, x × x = 21025

⇒ x 2 = 21025

⇒ x = √21025 = 145

∴ there are 145 rows of chairs.

Question 9:

Find the smallest number that should be added to 1290 to make it a perfect square.

Now ,1225 = 35 2 < 1290 < 36 2 = 1296

We see that, 1290 + 6 = 1296 = 36 2

Thus, 6 is the smallest number which must be added.

Question 10:

Find a Pythagorean triplet whose one number is 10.

We know that for m being a natural number = (m 2 – 1, 2m, m 2 + 1) is a Pythagorean triplet.

Let 2m = 10

Then, m 2 – 1 = 24 and m 2 + 1 = 26

Also, 26 2 = 24 2. + 10 2

∴ (10, 24, 26) is the Pythagorean triplet.

Find more square root questions for practice.

Also Watch:

Practice Questions on Squares and Square Roots

1. Identify which of the following is not a perfect square.

2. Find the Pythagorean triplet, whose one member is 6.

3. Find the square root of the following using the prime factorisation method.

4. Find the square of the following using the long division method.

(i) 2030625

(iii) 34404

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• Mathematics

How to Solve Square Root Problems

Last Updated: March 7, 2023 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 372,272 times.

While the intimidating sight of a square root symbol may make the mathematically-challenged cringe, square root problems are not as hard to solve as they may first seem. Simple square root problems can often be solved as easily as basic multiplication and division problems. More complex square root problems, on the other hand, can require some work, but with the right approach, even these can be easy. Start practicing square root problems today to learn this radical new math skill!

Understanding Squares and Square Roots

• Try squaring a few more numbers on your own to test this concept out. Remember, squaring a number is just multiplying it by itself. You can even do this for negative numbers. If you do, the answer will always be positive. For example, (-8) 2 = -8 × -8 = 64 .

• 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 .

• On the other hand, numbers that don't give whole numbers when you take their square roots are called imperfect squares . When you take one of these numbers' square roots, you usually get a decimal or fraction. Sometimes, the decimals involved can be quite messy. For instance, √(13) = 3.605551275464...

• 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 .

• Note that although imaginary numbers can't be represented with ordinary digits, they can still be treated like ordinary numbers in many ways. For instance, the square roots of negative numbers can be squared to give those negative numbers, just like any other square root. For example, i 2 = -1

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.

• In our example, we would divide 6.45 into pairs like this: 6-.45-00 . Note that there is a "leftover" digit on the left — this is OK.

• In our example, the first group in 6-.45-00 is 6. The biggest number that is less than or equal to 6 when squared is 2 — 2 2 = 4. Write a "2" above the 6 under the radical.

• In our example, we would start by taking the double of 2, the first digit of our answer. 2 × 2 = 4. Next, we would subtract 4 from 6 (our first "group"), getting 2 as our answer. Next, we would drop down the next group (45) to get 245. Finally, we would write 4 once more to the left, leaving a small space to add onto the end, like this: 4_.

• In our example, we want to find the number to fill in the blank in 4_ × _ that makes the answer as large as possible but still less than or equal to 245. In this case, the answer is 5 . 45 × 5 = 225, while 46 × 6 = 276.

• Continuing from our example, we would subtract 225 from 245 to get 20. Next, we would drop down the next pair of digits, 00, to make 2000. Doubling the numbers above the radical sign, we get 25 × 2 = 50. Solving for the blank in 50_ × _ =/< 2,000, we get 3 . At this point, we have "253" above the radical sign — repeating this process once again, we get a 9 as our next digit.

• In our example, the number under the radical sign is 6.45, so we would simply slide the point up and place it between the 2 and 5 digits of our answer, giving us 2.539 .

Quickly Estimating Imperfect Squares

• For example, let's say we need to find the square root of 40. Since we've memorized our perfect squares, we can say that 40 is in between 6 2 and 7 2 , or 36 and 49. Since 40 is greater than 6 2 , its square root will be greater than 6, and since it is less than 7 2 , its square root will be less than 7. 40 is a little closer to 36 than it is to 49, so the answer will probably be a little closer to 6. In the next few steps, we'll narrow our answer down.

• In our example problem, a reasonable estimate for the square root of 40 might be 6.4 , since we know from above that the answer is probably a little closer to 6 than it is to 7.

• 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.

• In our example, let's pick 6.33 for our two-decimal point estimate. Multiply 6.33 by itself to get 6.33 × 6.33 = 40.0689. Since this is slightly above our original number, we'll try a slightly lower number, like 6.32. 6.32 × 6.32 = 39.9424. This is slightly below our original number, so we know that the exact square root is between 6.33 and 6.32 . If we wanted to continue, we would keep using this same approach to get an answer that's continually more and more accurate.

Calculator, Practice Problems, and Answers

Expert Q&A

• For quick solutions, use a calculator. Most modern calculators can instantly find square roots. Usually, all you need to do is to simply type in your number, then press the button with the square root symbol. To find the square root of 841, for example, you might press: 8, 4, 1, (√) and get an answer of 29 . [16] X Research source Thanks Helpful 0 Not Helpful 1

Video . By using this service, some information may be shared with YouTube.

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

About This Article

To solve square root problems, understand that you are finding the number that, when multiplied by itself, equals the number in the square root. For quick recall, memorize the first 10-12 perfect squares, so that you recognize the square root of numbers like 9, 25, 49, or 121. If possible, break the number under the square root into individual perfect squares. For example, √(900) can be broken into √(9) × √(100), and √(100) can be broken into √(25) × √(4), reducing the problem to √(9) × √(25) × √(4), or 3 x 5 x 2 for an answer of 30. If you want to learn how to estimate imperfect square roots, keep reading the article! Did this summary help you? Yes No

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• Square Root Questions

What is Square Root?

A square root is a value that, when multiplied by itself, gives the original number.

How to Find the Square Root?

We can use two methods to find square roots - Prime factorization and the Long division method.

What are Squares and Square Roots?

Just the opposite method of squaring a number is the square root. If a number n, such as n 2 , is squared, then the square root of n 2 is equal to the original number n.

How to Square a Number?

We need to multiply the number by itself to find the square of a number.

For example, 3 multiplied by 3 is equal to 9.

Square of 4 is: 4 multiplied by 4 = 4 × 4 = 16.

Square Root Problems and Answers

Q1: Can you find out the number of numbers lying between the squares of these following pairs of numbers?

(i) 25 and 26

(ii) 99 and 100

Solution - As we know, between n 2 and (n + 1) 2 , the number of non–perfect square numbers are 2n.

(i) Between 252 and 262 there are 2 × 25 = 50 natural numbers.

(ii) Between 992 and 1002 there are 2 × 99 = 198 natural numbers.

Q2: Write a Pythagorean triplet whose one of the required member is:

We know, for any natural number m, 2m, m 2 – 1, m 2 + 1 is a Pythagorean triplet.

⇒ m = 6/2 = 3

m 2 – 1 = 3 2 – 1 = 9 – 1 = 8

m 2 + 1 = 3 2 + 1 = 9 + 1 = 10

Therefore, (6, 8, 10) is a Pythagorean triplet.

(ii) 2m = 14

⇒ m = 14/2 = 7

m 2 – 1 = 72 – 1 = 49 – 1 = 48

m 2 + 1 = 72 + 1 = 49 + 1 = 50

(14, 48, 50) is not a Pythagorean triplet.

(iii) 2m = 16

⇒ m = 16/2 = 8

m 2 – 1 = 8 2 – 1 = 64 – 1 = 63

m 2 + 1 = 8 2 + 1 = 64 + 1 = 65

Therefore, (16, 63, 65) is a Pythagorean triplet.

(iv) 2m = 18

⇒ m = 18/2 = 9

m 2 – 1 = 92 – 1 = 81 – 1 = 80

m 2 + 1 = 92 + 1 = 81 + 1 = 82

Therefore, (18, 80, 82) is a Pythagorean triplet.

Q3: (n + 1) 2 - n 2 = ?

(n + 1) 2 - n 2

= (n 2 + 2n + 1) – n 2

Q4: State that the number 121 is the sum of 11 odd natural numbers.

Solution - As 121 = 112

We recognize that n2 is the sum of the first n odd natural numbers.

It shows that 121 = the sum of the first 11 odd natural numbers.

= 1 + 3 + 5+ 7 + 9 + 11 +13 + 15 + 17 + 19 + 21

Q5: Use the identity and find the square of 189.

(a – b) 2 = a 2 – 2ab + b 2

Solution - 189 = (200 – 11) 2

= 40000 – 2 × 200 × 11 + 112

= 40000 – 4400 + 121

Q6: Find out the square root of 625 using the mathematical identity as stated:  (a + b) 2 = a 2 + b 2 + 2ab?

Solution - (625) 2

= (600 + 25) 2

= 6002 + 2 × 600 × 25 + 252

= 360000 + 30000 + 625

Properties of Square Roots

A square root function is defined in mathematics as a one-to-one function that takes as an input a positive number and returns the square root of the given input number.

The following are some of the essential properties of the square root:

If a number is a perfect square number, a perfect square root exists.

It may have a square root if a number ends with an even number of zeros (0's).

It is possible to multiply the two square root values. For instance, √3 can be multiplied by √2, then √6 should be the result.

If two same square roots are multiplied, then a radical number should be the product. This implies that a non-square root number is a product. When √7 is multiplied by √7, for example, the result obtained is 7.

It does not define the square root of any negative numbers. And no negative can be the perfect square.

If a number ends with 2, 3, 7, or 8 (in the digit of the unit), the perfect square root will not exist.

If in the unit digit, a number ends with 1, 4, 5, 6, or 9, the number would have a square root.

Using the prime factorization process, the square root of a perfect square number is simple to measure. For example -

Square Root By Prime Factorisation

With the help of an example, let us understand this concept:

Example 1: Solve √10 to 2 decimal places.

Step 1: Choose any two perfect square roots between which you feel your number may fall.

We know that 22 = 4; 32 = 9, 42 = 16 and 52 = 25

Now, choose 3 and 4 (as √10 lies between these 2 numbers)

Step 2: Divide the given number into one of the square roots chosen.

Divide 10 by 3.

=> 10/3 = 3.33 (round off answer at 2 places)

Step 3: Find the root average and the product of the step above, i.e.

(3 + 3.33)/2 = 3.1667

Verify: 3.1667 × 3.1667 = 10.0279 (Not required)

Repeat step 2 and step 3

Now 10/3.1667 = 3.1579

Average of 3.1667 and 3.1579.

(3.1667+3.1579)/2 = 3.1623

Verify: 3.1623 × 3.1623 = 10.0001 (more accurate)

Stop the process.

Example 2: Find the square roots of whole numbers from 1 to 100 that are perfect squares.

Solution - The perfect squares are - 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Example 3: What is:

The square root of 2.

The square root of 3.

The square root of 4.

The square root of 5

Solution - Use a square root list, we have

Value of root 2 i.e. √2 = 1.4142.

Value of root 3 i.e. √3 = 1.7321.

Value of root 4 i.e. √4 = 2.

Value of root 5 i.e. √5 = 2.2361.

FAQs on Square Root Questions

Question 1. Define Square Numbers and Square Roots.

Answer: Square numbers can be defined as the numbers that are produced when a number is multiplied by itself. For example, if n is a number and it is multiplied by itself, then the square of n can be given as n 2 . Another example is the square of 10, which is 10 2 = 10 x 10 = 100.

Further, the square root of a number can be explained as the value which when multiplied by itself gives the original number. This value can be represented by the symbol ‘√.’ For example, the square root of 25 is √25 = 5.

Question 2. What do you Understand by Perfect Squares? Provide Examples with Your Answer.

Answer: Perfect squares can be defined as numbers in which the square root of the number provides a whole number. For example, 9 is a perfect square number because its root is a whole number, which means that √9 = 3.

Question 3. What is an Imperfect Square? Provide Examples with Your Answer.

Answer: An imperfect square can be defined as a number in which the square root of the number is a fraction. The value is generated by taking the square root of the imperfect square can also be non-terminating. For example, 3 is an imperfect square because the root of this number is 1.73205080757. This number is a fraction.

Question 4. What is the Main Difference Between a Square and Square Roots?

Answer: The square root of a number provides the root of a particular number that was squared. This is the main difference between a square number and square roots.

Question 5. How to Solve the Square Root Equation.

Answer: We need to follow the below steps to solve the square root equation: Isolate the square to one of the sides (L.H.S or R.H.S). Now solve the remaining equation. Square both sides of the given equation. With examples, let us understand the steps.

Question 6. Is the Square Root of a Negative Number a Whole Number?

Answer: No, negative numbers shouldn't have a square root, as per the square root concept. If we multiply two negative numbers, a positive number will always be the product. Negative number square roots expressed as multiples of I (imaginary numbers).

Aptitude - Square Root and Cube Root

Why should i learn to solve aptitude questions and answers section on "square root and cube root".

Learn and practise solving Aptitude questions and answers section on "Square Root and Cube Root" to enhance your skills so that you can clear interviews, competitive examinations, and various entrance tests (CAT, GATE, GRE, MAT, bank exams, railway exams, etc.) with full confidence.

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IndiaBIX provides you with numerous Aptitude questions and answers based on "Square Root and Cube Root" along with fully solved examples and detailed explanations that will be easy to understand.

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Here you can find multiple-choice Aptitude questions and answers based on "Square Root and Cube Root" for your placement interviews and competitive exams. Objective-type and true-or-false-type questions are given too.

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• Square Root and Cube Root - Formulas
• Square Root and Cube Root - General Questions

   = 0.06

Then x 2 = 128 x 162

   = 64 x 2 x 18 x 9

   = 8 2 x 6 2 x 3 2

   = 8 x 6 x 3

   = 144.

L.C.M. of 21, 36, 66 = 2772.

Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11

To make it a perfect square, it must be multiplied by 7 x 11.

So, required number = 2 2 x 3 2 x 7 2 x 11 2 = 213444

If 3 5 + 125 = 17.88, then what will be the value of 80 + 6 5 ?

3 5 + 125 = 17.88

   = 4 5 + 6 5

   = 10 5 = (10 x 2.235) = 22.35

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Square Root Questions

Square and square root practice test:.

The largest of four digit numbers which is a perfect square is (a) 9801 (b) 9904 (c) 9804 (d) 9809 View Answer

Evaluate √6084 by factorization method . (a) 78 (b) 76 (c) 82 (d) 64 View Answer

Related: number series questions

What is the square root of 8? (a) 2.828 (b) 2.928 (c) 2.728 (d) 2.628 View Answer

Find the square root of each of 291600 by the long division method: (a) 520 (b) 620 (c) 580 (d) 540 View Answer

What is the square root of 0.0009? (a) 0.81 (b) 0.27 (c) 0.03 (d) 0.003 View Answer

Find the 169 square root. (a) 21 (b) 11 (c) 17 (d) 13 View Answer

Related: Fruit Trivia Questions (and Answers)

Using the square root ability, find the value 6.328 (a) 2.512 (b) 2.518 (c) 2.517 (d) 2.514 View Answer

What is the square root of 85? (a) 9.519 (b) 9.419 (c) 9.319 (d) 9.219 View Answer

Evaluate: √0.9 up to 3 places of decimal. (a) 0.948 (b) 0.958 (c) 0.938 (d) 0.978 View Answer

A general arranges his soldiers in a row to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers is 8160, find the number of soldiers in each row. (a) 96 soldiers (b) 92 soldiers (c) 90 soldiers (d) 99 soldiers View Answer

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Square Root

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If 'a' is the square root of 'b', it means that a × a = b. The square of any number is always a positive number, so every number has two square roots, one of a positive value, and one of a negative value. For example, both 2 and -2 are square roots of 4. However, in most places, only the positive value is written as the square root of a number.

What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents . Consider the number 9. When 3 is multiplied by itself, it gives 9 as the product. This can be written as 3 × 3 or 3 2 . Here, the exponent is 2, and we call it a square. Now when the exponent is 1/2, it refers to the square root of the number. For example, √n = n 1/2 , where n is a positive integer.

Square Root Definition

The square root of a number is the value of power 1/2 of that number. In other words, it is the number whose product by itself gives the original number. It is represented using the symbol '√ '. The square root symbol is called a radical , whereas the number under the square root symbol is called the radicand.

How to Find Square Root?

To find the square root of a number, we just see by squaring which number would give the actual number. It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that can be expressed as the product of a number by itself. In other words, perfect squares are numbers which are expressed as the value of power 2 of any integer . We can use four methods to find the square root of numbers and those methods are as follows:

• Repeated Subtraction Method
• Prime Factorization Method
• Estimation Method
• Long Division Method

It should be noted that the first three methods can be conveniently used for perfect squares, while the fourth method, i.e., the long division method can be used for any number whether it is a perfect square or not.

Repeated Subtraction Method of Square Root

This is a very simple method. We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of 16 using this method.

• 16 - 1 = 15

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

Square Root by Prime Factorization Method

Prime factorization of any number means to represent that number as a product of prime numbers . To find the square root of a given number through the prime factorization method, we follow the steps given below:

• Step 1: Divide the given number into its prime factors .
• Step 2: Form pairs of 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.

Finding Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. Let us use this method to find √15. Find the nearest perfect square number to 15. 9 and 16 are the perfect square numbers nearest to 15. We know that √16 = 4 and √9 = 3. This implies that √15 lies between 3 and 4. Now, we need to see if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Since 3.5 2 = 12.25 and 4 2 = 16. Thus, √15 lies between 3.5 and 4 and is closer to 4.

Let us find the squares of 3.8 and 3.9. Since 3.8 2 = 14.44 and 3.9 2 = 15.21. This implies that √15 lies between 3.8 and 3.9. We can repeat the process and check between 3.85 and 3.9. We can observe that √15 = 3.872.

This is a very long process and time-consuming.

Calculating Square Root by Long Division Method

Long division is a method for dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. We can find the exact square root of any given number using this method. Let us understand the process of finding square root by the long division method with an example. Let us find the square root of 180.

• 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.

Step 3: Bring down the number under the next bar to the right of the remainder . Add the last digit of the quotient to the divisor . To the right of the obtained sum, find a suitable number which, together with the result of the sum, forms a new divisor for the new dividend that is carried down.

Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same — as being either less than or equal to the dividend.

Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.

Step 6: The quotient thus obtained will be the square root of the number. Here, the square root of 180 is approximately equal to 13.4 and more digits after the decimal point can be obtained by repeating the same process as follows.

Square Root Table

The square root table consists of numbers and their square roots. It is useful to find the squares of numbers as well. Here is the list of square roots of perfect square numbers and some non-perfect square numbers from 1 to 10.

The numbers that are not perfect squares are irrational numbers .

Square Root Formula

The square root of a number has the exponent of 1/2. The square root formula is used to find the square root of a number. We know the exponent formula : \(\sqrt[\text{n}]{x}\) = x 1/n . When n = 2, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, and so on. 9 1/2 = √9 = √(3×3) = 3. So, the formula for writing the square root of a number is √x= x 1/2 .

Simplifying Square Root

To simplify a square root, we need to find the prime factorization of the given number. If a factor doesn't have a pair, retain them under the square root symbol, otherwise, take one number out of the square root from each pair. For example: √12 = \(\sqrt{2 \times 2\times3}\) = 2√3. This is because the rule of simplifying square root is √xy = √(x × y), where, x and y are positive integers.

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

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. The principal square root of -x is: √(-x)= i√x. Here, i is the square root of -1.

For example: Take a perfect square number like 16. Now, let's see the square root of -16. There is no real square root of -16. √(-16)= √16 × √(-1) = 4i (as, √(-1)= i), where 'i' is represented as the square root of -1. So, 4i is a square root of -16.

Square of a Number

Any number raised to exponent two (y 2 ) is called the square of the base. So, 5 2 or 25 is referred to as the square of 5, while 8 2 or 64 is referred to as the square of 8. We can easily find the square of a number by multiplying the number two times. For example, 5 2 = 5 × 5 = 25, and 8 2 = 8 × 8 = 64. When we find the square of a whole number, the resultant number is a perfect square. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number is always a positive number.

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How to Find the Square of a Number?

The square of a number can be found by multiplying a number by itself. For single-digit numbers, we can use multiplication tables to find the square, while in the case of two or more than two-digit numbers, we perform multiplication of the number by itself to get the answer. For example, 9 × 9 = 81, where 81 is the square of 9. Similarly, 3 × 3 = 9, where 9 is the square of 3.

The square of a number is written by raising the exponent to 2. For example, the square of 3 is written as 3 2 and is read as "3 squared". Here are some examples:

• 4 2 = 4 × 4 = 16
• (-6) 2 = -6 × -6 = 36
• (5/3) 2 = 5/3 × 5/3 = 25/9

Squares and Square Roots

There is very strong relation between squares and square roots as each one of them is the inverse relation of the other. i.e., if x 2 = y then x = √y. It can be simply remembered like this:

• 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.

Squaring on both sides of the equation would result in the cancellation of the square root on the left side.

2x + 3 = 10 2

2x + 3 = 100

x = 97/2 = 48.5

Here are more differences between squares and square roots.

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Examples on Square Root

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

Prime factorization of 529 .

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

∴ √529= 23.

Example 2: Calculate the square and square root of the following numbers.

a) Square root of 25 is ___

b) Square of 16 is ____

c) Square of 20 is ____

d) Square root of 400 is _____

a) Square root of 25 is 5

5 × 5 = 25

√25 = 5

b) Square of 16 is = 16 × 16 = 256

c) Square of 20 is = 20 × 20 = 400

d) Square root of 400 is 20 as

20 × 20 = 400

√400 = 20.

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

Example 3: Determine the square root of 60.

To find: the square root of 60

From prime factorization of 60 , we get,

60 = 2 × 2 × 3 × 5

= (2) 2 × 3 × 5

Using square root formula,

√60 = [(2) 2 × 15 ] 1/ 2

√60 = 2√15

Therefore, the square root of 60 = 2√15

Answer: 2√15

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Practice Questions on Square Root

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FAQs on Square Root

What is square root in math.

The square root of a number is a number that when multiplied by itself gives the actual number. For example, 2 is the square root of 4, and this is expressed as √4 = 2. This means when 2 is multiplied by 2 it results in 4 and this can be verified as 2 × 2 = 4.

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How to Calculate the Square Root of a Number?

It is very easy to find the square root of a number that is a perfect square . For example, 9 is a perfect square, 9 = 3 × 3. So, 3 is the square root of 9 and this can be expressed as √9 = 3. The square root of any number, in general, can be found by using any of the four methods given below:

• Estimation and Approximation Method

Can Square Root be Negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like 4, 9, 25, 36, etc. have two square roots, one is a positive value and one is a negative value. For example, the square roots of 4 are -2 and 2. To verify this, we can see that (-2) × (-2) = 4. Similarly, the square roots of 9 are 3 and -3.

How to Find the Square Root of a Decimal Number?

The square root of a decimal number can be found by using the estimation method or the long division method. In the case of decimal numbers, we make pairs of whole number parts and fractional parts separately. And then, we carry out the process of long division in the same way as any other whole number.

What is the Square Root Symbol?

The symbol that is used to denote square root is called the radical sign '√ '. The term written inside the radical sign is called the radicand.

How to Multiply Two Square Root Values Together?

Let us say we have two numbers a and b. 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 understand this with a practical illustration. For example, multiply √4 × √16. The square root of 4 is 2 (√4 = 2) and the square root of 16 is 4 (√16 = 4). Now, we will multiply the value of the square root of 4 and 16, i.e., 2 × 4 = 8. Instead, we can apply the property of square roots, √a × √b = √ab.

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

The square root of any number can be expressed using the formula: √y = y ½ . In other words, if a number has 1/2 as its exponent, it means we need to find the square root of the number.

What is the Square and Square Root of a Number?

The square of a number is the product that we get on multiplying a number by itself. For example, 6 × 6 = 36. Here, 36 is the square of 6. The square root of a number is that factor of the number and when it is multiplied by itself the result is the original number. Now, if we want to find the square root of 36, that is, √36, we get the answer as, √36 = 6. Hence, we can see that the square and the square root of a number are inverse operations of each other.

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

In Math, a non-perfect or an imperfect square number is considered as a number whose square root cannot be found as an integer or as a fraction of integers. The square root of a non-perfect square number can be calculated by using the long division method.

How to Find a Square Root on a Calculator?

To find the square root value of any number on a calculator, we simply need to type the number for which we want the square root and then insert the square root symbol √ in the calculator. For example, if we need to find the square root of 81, we should type 81 in the calculator and then press the symbol √ to get its square root. We will get √81 = 9.

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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?

The product that we get on multiplying a number by itself is the square of the number. For example, 5 × 5 = 25. Here, 25 is the square of 5 and this can also be written as 5 2 = 25.

How to Calculate the Square Root of a Negative Number?

Note that the square root of a negative number is not a real number . It is an imaginary number . For example, √(-4) = √(-1) × √4 = i (2) = 2i, where 'i' is known as " iota " and i 2 = -1 (or) i = √(-1).

Why is the Square of a Negative Number Positive?

The square of a negative number is positive because when two negative numbers are multiplied it always results in a positive number. For example, (-4) × (-4) = 16.