Square Root : Definition, Formula, Methods and Types Explained

Square Root

Square roots are one of those seemingly daunting maths topics that appear in many different situations, from algebra to geometry. Yet the concepts behind them aren’t as hard to grasp. It makes handling numbers far easier if you know the concept well. Let us understand how to find the square roots of a number here.

What are square roots?

A square root is the number that you multiply by itself and get the original number. For example, 4 × 4 = 16. Hence, the square root of 16 equals 4. Knowing what are square roots provides a strong base for advanced concepts.

Square and square root definition

The square and square root definition is based on inverse operations.

• Squaring: 8² = 8 × 8 = 64
• Square root: √64 = 8

Squaring builds the number. Finding the square root brings it back. These two processes are opposites of each other.

How to calculate a square root?

There is no single way of how to calculate a square root value. The method depends on the type of number.

Repeated Subtraction Method for Square Root

This method uses a pattern. Perfect squares can be reduced to zero by subtracting consecutive odd numbers.

Example with 16:

16 – 1 = 15

15 – 3 = 12

12 – 5 = 7

7 – 7 = 0

Four steps were needed, so √16 = 4.

It is not the fastest method, but it helps students notice number patterns.

Square Root by Prime Factorization Method

This method for square root breaks numbers into prime factors. Here is how to calculate:

• Find the prime factors of the number
• Group factors into pairs
• Take one number from each pair
• Multiply those numbers

Here is an example:

100 = 2 × 2 × 5 × 5

Group identical factors:

(2 × 2) and (5 × 5)

Take one from each pair:

√100 = 2 × 5 = 10

This method for square root is reliable and widely used in schools.

Finding Square Root by Estimation Method

Not every number is a perfect square. For example, 20.

4² = 16

5² = 25

So √20 lies between 4 and 5. With closer checking, it is about 4.47.

Estimation helps when exact values are not necessary.

Calculating Square Root by Long Division Method

The long division method looks lengthy, but it gives accurate results.

Digits are grouped in pairs from right to left. Then, the largest square smaller than the first group is found. The process continues step by step.

Though it requires practice, it works for both perfect and non-perfect squares.

Squares and square roots table

A squares and square roots table helps in quick recall.

Number	Square	Square Root
1	1	1
2	4	2
3	9	3
4	16	4
5	25	5
6	36	6
7	49	7
8	64	8
9	81	9
10	100	10
11	121	11
12	144	12

Memorising this makes many problems easier.

Formula for Square Root

The formula for square root is written as:

√x = x^^(1/2)

This shows that a square root is the same as raising a number to the power one-half.

Simplifying Square Root

Some square roots cannot be written as whole numbers, but they can be simplified.

Example:

√45

45 = 9 × 5

√45 = √9 × √5

= 3√5

Simplifying makes expressions cleaner.

Square Root of a Negative Number

In real numbers, √(–4) does not exist. No real number multiplied by itself gives a negative result.

In higher mathematics, √(–4) is written as 2i, where i² = –1. These are imaginary numbers.

Square of a Number

The square of a number means multiplying it by itself.

For example:

12² = 144

Squaring is common in area calculations and algebraic identities.

How to Find the Square of a Number?

To find the square:

• Multiply the number by itself.
• Use algebraic identities when needed.
• Apply shortcuts for numbers ending in 5.

Example:

25² = 625

Practice improves speed.

Types of Square Roots

The types of square roots are divided into two broad categories:

• Perfect square roots: Numbers that produce whole-number answers, like √4 = 2, √9 = 3.
• Non-perfect square roots: These are the numbers whose square roots provide indefinite number results that are non-repeating. These are irrational numbers. Examples include √3 = 1.732…, √5 = 2.236…

Square Root of Numbers

The square root of numbers can be exact or approximate.

Examples:

√36 = 6

√8 ≈ 2.82

Whether to calculate square root exactly or approximately depends on the question.

Examples of Square Root

1. √121 = 11
2. √196 = 14
3. Simplify √32 = 4√2
4. Estimate √18

These examples cover different methods.

Practice Questions on Square Root

1. Find √225
2. Simplify √72
3. Calculate square root of 400
4. Calculate the square of 19
5. Estimate √50

Conclusion

Square roots may be trickier first off, especially when students see various methods for square root options at one time. But once they have the basic idea, the topic starts to feel easy. With enough practice and a good understanding of all the different ways to do square roots, confidence comes over time.

For students who require extra guidance, structured guidance and learning experience can make a difference. At Turito, students can enrol in online courses that cover concepts step by step and offer frequent practice. Square roots become much less confusing and far easier with adequate support. Visit our website and explore our 

FAQs