Squares and Square Roots

In math, squares and square roots are inverse operations. Square of a number is the value of power 2 of the number, while the square root of a number is the number that we need to multiply by itself to get 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 positive value, and one negative value. For example, both 2 and -2 are square roots of 4. But at most places, you will find that only the positive value is written as the square root.

Table of Contents

Square of a Number

Any number raised to the exponent two is called the square of the base. So, 52 is referred to as the square of 5, while 82 is referred to as the square of 8. We can easily find the square of a number by multiplying the base two times. For example, 5 squared is 5×5=25, and 8 squared is 8×8=64. When we find the square of a whole number, the resultant number is one of the perfect squares. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number, whether it is positive or negative, is always a positive number.

How to Find Square?

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 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. 

squares and square roots

What is Square Root?

The square root of a number is the number that gets multiplied to itself to give the product. We have learned about exponents. Squares and square roots are special exponents. Consider the number 9. Which number, when multiplied by itself, gives 9 as the product? When the exponent is 2, it is called a square. When the exponent is 1/2, it is called a square root. For example √(n×n)=√n2= n, where n is a positive integer.

Square Root: Definition

Square root of a number is the value of power 1/2 of that number. In other words, it is the number that we multiply by itself to get the original number. It is represented using the symbol '√ '.

How to find Square Root?

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 written as the multiplication of a number by itself. in other words, perfect squares are numbers which is 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 of Square Root
  • Square Root by Prime Factorization Method
  • Square Root by Estimation Method
  • Square Root by Long Division Method.

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

Repeated Subtraction Method of Square Root

This is a very simple method. We will 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. It works only for perfect square numbers. Let us find the square root of 16 using this method.

  • 16- 1= 15
  • 15- 3=12
  • 12- 5=7
  • 7- 7= 0

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 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.

Square root of 144 

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

Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and 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 numbers 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. 3.52 = 12.25 and 42= 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. 3.82 = 14.44 and 3.92 = 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.

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's find the square root of 180.

  • Step 1: Place a bar over every pair of digits of the number starting from the unit’s 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 by long division method

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.

square root by long division method

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.

square root by long division method

Step 6: The quotient thus obtained will be the square root of the number.

Square Root Table

Square root table comprises numbers and their square roots. It is useful to find 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.

Number Square Root
1 1
2 1.414
3 1.732
4 2
5 2.236
6 2.449
7 2.646
8 2.828
9 3
10 3.162

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

Square Root Formula

The square root is nothing but the exponent 1/2. Square root formula is used to find the square root of a number. We know the exponent formula: \(\sqrt[\text{n}]{x}\) = x1/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, long division, and so on. 91/2 = √9 = √(3×3) = 3. So, the formula for writing the square root of a number is √x= x1/2.

How to simplify Square Root?

To simplify a square root, we need to find the prime factorization of the given number. If a factor cannot be grouped, retain them under the square root symbol. The rule of simplifying square root is: √xy = √(x × y), where, x and y are positive integers. For example: √12 = \(\sqrt{2 \times 2\times3}\) = 2√3                  

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 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.


FAQs on Square Root

What is a Square Root?

Square root is the number that we need to multiply by itself to get the original number. For example, 2 is the square root of 4, as 2×2= 4.

How to Find Square Root?

Square root of a number can be found by using any of the four methods given below:

  • Repeated Subtraction Method
  • Prime Factorization Method
  • Estimation and Approximation Method
  • Long Division Method.

How to Find Square Root of Decimal Number?

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

Can Square Root be Negative?

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 positive value and one negative value. Square roots of 4 are -2 and 2. Similarly, square roots of 9 are 3 and -3.

What do you Call the Square Root Symbol?

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

What is the Formula of Square Root?

The square root of any number can be expressed using the formula: √x= x1/2.


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Solved Examples

Example 1: Help Mary find the square root of 169 by repeated subtraction method.

Solution: 

  • 169 - 1= 168
  • 168 - 3 = 165
  • 165 - 5 = 160
  • 160 - 7 = 153
  • 153 - 9 = 144
  • 144 - 11 = 133
  • 133 - 13 = 120
  • 120 - 15 = 105
  • 105 - 17 = 88
  • 88 - 19 = 69
  • 69 - 21 = 48
  • 48 - 23 = 25
  • 25 - 25 = 0

We have done repeated subtraction 13 times. Therefore, √169 = 13.

Example 2: Help Kate find out the square root of 529 by the prime factorization method.

Solution:

Prime factorization of 529.

square root of 529

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

∴ √529= 23.

Practice Questions on Square Root

Here are a few problems for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
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