Square Root Tricks with Examples and FAQs

Square Root Tricks

Before discussing the steps to make finding square roots easier, let’s first define what are square roots.

Square Roots

A number when multiplied to itself gives a number which is its square and the number itself is called the square root of the resulting number.

The symbol to represent root is √ .  The square root of a whole number can be a rational or irrational number. The numbers whose roots are a whole number are called perfect squares.

The square roots of a number can have both positive and negative values, in this article we will discuss the tricks to find the positive roots. 

For example,

7 × 7 = 49 ⇒ 49 is the square of 7. 

(-7) × (-7) = 49 ⇒ 49 is the square of (-7).

49 = ±7   ⇒  ±7 is the square root of 49.

We can also say that 49 is a perfect square.

Table of Squares of first 10 Natural Numbers

Table of Square Root of first 10 Natural numbers

Trick

The trick to find the square root of numbers that are greater than 100 are:

1. First, start grouping up the digits from right to left, first in pair of two and the remaining another group.

For example,

If the number is a 3 digit number, let the number be 324: grouping is 3 24 

If the number is a 4 digit number, let the number be 1764: grouping is 17 64 

If the number is a 5 digit number, let the number be 12544: grouping is 125 44 

2. The unit digit of the first pair from the right, is used to get an idea of the resulting numbers unit digit from the table below. Where the squares of the first 10 natural numbers are given with the unit’s digit of each square.

For example,

If the number is 324 and its grouping is 3 24, now in the first pair from right we see that units digit is 4. We check from the table that the squares of 2 and 8 have 4 at their unit’s place. So, the units place of the square root of 324 will be 2 or 8.

3. The second group of the number is taken, and then we need to find two numbers from the list between the squares of which this number lies. The smaller of the two numbers is the ten’s digit of the square root.

For example,

If the number is 324 and its grouping is 3 24, now from the second part we see that the number is 3. 3 lies between the squares of 1 and 2.

1^{2}<3<2^{2}

Now as 1 < 2, hence the tens digit of the required square root is 1.

Now the square root of 324 can be either 12 or 18.

4. The product of the probable digits which can be at the tens place is taken. If the product value is greater than the second group from the right, the units digit of the square root is lesser of the two options available for the units’ digit.

And if it is lesser than or equal to the second part from the right it is the greater of the two options.

In the example, we have been discussing already,

1 × 2 = 2,

2 ≤ 3,

Hence the units digit is greater of the digits among 2 and 8,

The following examples will make understanding the trick easier.

Examples

Square root of a 3-digit Number

Find \sqrt{576}.

Pairing the digits from right to left, 

5 76

The unit digit of the number is 6 hence the unit digit of the square root will be 4 or 6.

Now the number 5 lies between 

2^{2}<5<3^{2}

Hence the tens digit of the square root of 576 is 2 as 2 < 3.

Therefore, the square root is either 24 or 26.

Taking the products of possible tens digits 2 × 3 = 6,

As 6 > 5, the units digit is the lesser of the two, among 4 and 6.

\sqrt{576}=24

Square Root of a 4-digit Number

Pairing the digits from right to left, 

18 49

The unit digit of the number is 9 hence the unit digit of the square root will be 3 or 7.

Now the number 18 lies between 

4^{2}<18<5^{2}

Hence the tens digit of the square root of 1849 is 4 as 4 < 5.

Therefore, the square root is either 43 or 47.

Taking the products of possible tens digits 4 × 5 = 20,

As 20 > 18, the units digit is the lesser of the two, among 3 and 7.

\sqrt{1849}=43

Square Root of a 5-digit Number

\text { Find } \sqrt{28224}.

Pairing the digits from right to left, 

282 24

The unit digit of the number is 4 hence the unit digit of the square root will be 2 or 8.

Now the number 262 lies between 

16^{2}<282<17^{2}

Hence the first two digits of the square root of 28224 are 16 as 16 < 17.

Therefore, the square root is either 162 or 168.

Taking the products of possible first two digits 16 × 17 = 272,

As 272 282, the units digit is the greater of the two, among 2 and 8.

\sqrt{28224}=168