How To Find Square Root Of A Number with Solved Examples

The square root of a number is the inverse of squaring a number. The square of a number is the value acquired by multiplying the number by itself, whereas the square root of a number is obtained by finding a number that, when squared, equals the original number.

If ‘a’ is the square root of ‘b’, then a × a = b. The square of every integer is always positive, hence every number has two square roots, one positive and one negative. For example, both 2 and -2 are square roots of four. However, in most cases, only the positive value is expressed as the square root of a number.

Square Root Definition

The square root of any number equals a number that, when squared, gives the original value.

Let m be a positive integer, such that √(m.m) = √(m²) = m

In mathematics, a square root function is a one-to-one function that takes a positive number as input and returns its square root.

f(x) = √x

For example, if x = 9, then the function returns the output value as 3.

Also Read: Properties of Rational Numbers

Square Root Symbol

The square root sign is typically represented as ‘√’. It is known as a radical symbol. To express a number ‘x’ as a square root, use this symbol:’√x’.

where x represents the number. The number under the radical symbol is known as the radicand. For example, the square root of 6 is also known as the radical of 6. Both express the same value.

How To Find Square Root Of A Number

To obtain the square root of a number, simply squaring the value yields the real number. Finding the square root of a perfect square number is a simple task. Perfect squares are positive numbers that can be stated as a number multiplied by itself. In other terms, perfect squares are integers that represent the power 2 of any integer. We can use four methods to find the square root of numbers, which are as follows:

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

It should be emphasised that the first three ways are more useful for perfect squares, however the fourth method, i.e. the long division method, can be applied to any number, perfect square or not.

Also Read: Commutative Property

Square Root by Repeated Subtraction Method

This is an extremely basic way. We remove consecutive odd numbers from the number for which we are calculating the square root until we reach zero. The number of times we subtract equals the square root of the provided integer. This approach only works with perfect square integers. Let’s use this method to find the square root of 16.

16 – 1 = 15

15 – 3 =12

12 – 5 = 7

7- 7 = 0

You can see that we’ve deducted four times. Thus,√16 = 4

Square Root by Prime Factorization Method

The square root of a perfect square number is simple to calculate using the prime factorization method. Let’s solve some of the examples here.

Number	Prime Factorization	Square Root
16	2x2x2x2	√16 = 2×2 = 4
144	2x2x2x2x3x3	√144 = 2x2x3 = 12
169	13×13	√169 = 13
256	256 = 2×2×2×2×2×2×2×2	√256 = (2x2x2x2) = 16
576	576 = 2x2x2x2x2x2x3x3	√576 = 2x2x2x3 = 24

Square Root by Estimation Method

This method is used as an approximation to calculate the square root by estimating the numbers.

For example, the square root of 4 is 2 and the square root of 9 is 3, therefore we may assume the square root of 5 will be between 2 and 3.

However, we need to determine whether the value of 5 is closer to 2 or 3. Let’s calculate the square of 2.2 and 2.8.

2.2² = 4.84

2.8² = 7.84

Since the square of 2.2 yields an approximate value of 5, we can estimate that the square root of 5 is roughly equal to 2.2.

Square Root by Long Division Method

Finding square roots for imperfect numbers is tricky, but we can do so using the long division approach. This can be understood using the example provided below. Consider the example of calculating the square root of 128.

1. Group the digits: Split 128 into {1}, {28}.
2. Start with the leftmost group ({1}): Find the largest perfect square less than or equal to it (1 in this case). Write 1 as the quotient (above the bracket) and 1 (1 squared) next to it.
3. Subtract and bring down: Subtract 1 (from step 2) from {1}, getting 0. Bring down {28} to form a new dividend, 028.
4. Double the quotient (1) and find the divisor: Write 2 (double of 1) next to the division sign. Beside 2, find the largest number (here, 2) that when multiplied by itself (2 x 2 = 4) is less than the new dividend (028). Write that number (2) next to 2, forming the divisor 22.
5. Multiply, subtract and bring down: Multiply the divisor (22) by the quotient digit (2) and write the result (44) next to the bracket. Subtract 44 from 028, getting 84. Write 84 below the line.
6. Stop (optional): Since 84 (remainder) is greater than the divisor squared (484), we’ve reached the maximum integer part of the square root. Further division would result in decimals.

Answer: The square root of 128, up to the integer part, is 12.

Square Root Table

The square root database contains numbers and their square roots. It is also handy for finding the squares of numbers. Here is a list of the square roots of perfect square numbers and certain non-perfect square numbers ranging from 1 to 50.

√n	Value	√n	Value	√n	Value
√1	1	√18	4.2426	√35	5.9161
√2	1.4142	√19	4.3589	√36	6
√3	1.7321	√20	4.4721	√37	6.0828
√4	2	√21	4.5826	√38	6.1644
√5	2.2361	√22	4.6904	√39	6.2450
√6	2.4495	√23	4.7958	√40	6.3246
√7	2.6458	√24	4.8990	√41	6.4031
√8	2.8284	√25	5	√42	6.4807
√9	3	√26	5.0990	√43	6.5574
√10	3.1623	√27	5.1962	√44	6.6332
√11	3.3166	√28	5.2915	√45	6.7082
√12	3.4641	√29	5.3852	√46	6.7823
√13	3.6056	√30	5.4772	√47	6.8557
√14	3.7417	√31	5.5678	√48	6.9282
√15	3.8730	√32	5.6569	√49	7
√16	4	√33	5.7446	√50	7.0711
√17	4.1231	√34	5.8310

Properties of Square Root

Some of the essential properties of the square root are listed below:

• If a number is perfect square, then it has a perfect square root.
• If a number contains an even number of zeros (0’s), it can have a square root.
• The two square root values may be multiplied. For instance, multiplying √3 by √2 should provide √6.
• When two identical square roots are multiplied, the result should be a radical number. It indicates that the result is not a square root number. For example, multiplying √7 by √7 yields a result of 7.
• The square root of any negative number does not exist since the perfect square cannot be negative.
• If a number ends in 2, 3, 7, or 8 (the unit digit), the perfect square root does not exist.
• If a number’s unit digit is 1, 4, 5, 6, or 9, it may have a perfect square root.

Also read: Area Of Isosceles Triangle with Solved Examples

Solved Examples

1. Calculate the square and square root of the following numbers.

a) Square root of 25 

b) Square of 16 

c) Square of 20 is 

Solution:

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

Answer: a) 5 b) 256 c) 400 

2. Find the square root of 60.

Solution:

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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