# Square Root and Cube Root

## Square Root

## A Number which produces a specified quantity when multiplied by itself.

A square root is a number, which is multiplied by itself to get the given number.

The sign of square root is √

We write square root of 3 as √3

Square root of 7 as √5 etc

# Cube Root

If a number x, which cube root is y if x = y^{3}

Then we can write 3√a = b

So the symbol of cube root is 3√

**Example :** 3√64 = 4, 3√125 = 5

Now we read examples which are related to square root and cube root.

## Methods to find square root

There are 2 methods to find square root.

- Prime factor method
- Division method

**(i) Prime factor method :**

In the method first we find prime factors of given number. Then we grouping the prime factors in 3-3 groups.

Examples : The square root of 125 by prime factor method

Sol :

\(\displaystyle \begin{array}{l}\underline{{5\left| {125} \right.}}\\\underline{{5\left| {25} \right.}}\\\underline{{5\left| 5 \right.}}\\\,\,\,\left| 1 \right.\end{array}\)

Prime Factors are : 5 × 5 × 5

= 5

So, the square root of 125 is 5.

### (ii) Division method :

In this method we set 2-2 groups from right side of given number.

Example : The square root of 625 division method**Step 1** Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.

Thus we have, \(\displaystyle \overline{6}\overline{{25}}\)

**Step 2** Find the largest number whose square is less than or equal to the number under the extreme left bar (22 < 6 < 32). Take this number as the divisor and the quotient with the number under the extreme left bar as the divided (here 6). Divide and get the remainder (2 in this case).

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,2\\\,\,\,\,\,\overline{{2\left| {\,\,\,\,625} \right.}}\\\left. {\,\,\,\,\,2} \right|\,\,-4\\\overline{{\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,2\,\,\,\,\,\,\,} \right.}}\end{array}\)

**Step 3** Bring down the number under the next bar (25 in this case) to the right of the remainder. So the new dividend is 225.

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,2\\\,\,\,\,\,\overline{{2\left| {\,\,\,\,625} \right.}}\\\left. {\,\,\,\,\,2} \right|\,\,-4\\\overline{{\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,225\,\,} \right.}}\end{array}\)

**Step 4** Double the quotient and enter it with a blank on its right.

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,2\\\,\,\,\,\,\overline{{2\left| {\,\,\,\,625} \right.}}\\\left. {\,\,\,\,\,2} \right|\,\,-4\\\overline{{\,\,4\underline{{\,\,}}\left| {\,\,\,\,\,\,225\,\,} \right.}}\end{array}\)

**substract the numbers**

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,} \right|25\\\overline{{2\,\,\,\left| {\,\,\,\underline{6}\underline{{25}}} \right.}}\\2\,\,\,\left| {-4} \right.\\\overline{{45\left| {\,\,\,225} \right.}}\\\underline{{\,\,\,5\left| {-225} \right.}}\\\,\,\,\,\,\,\left| {\,\,\,\,\,\times } \right.\end{array}\)

So, the square root of 625 is 25.

#### Others Examples

##### Q.1 Find the square rood of 17424 ?

Sol :

\(\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\left| {132} \right.\\\overline{{\,\,\,1\,\,\,\left| {\,\,\,\underline{1}\underline{{74}}\underline{{24}}} \right.}}\\\,\,\,1\,\,\,\left| {-1} \right.\\\overline{{\,\,23\left| {\,\,\,\,74} \right.}}\\\underline{{\,\,\,\,\,3\left| {\,-69} \right.}}\\262\left| {\,\,\,\,\,\,\,524} \right.\\\underline{{\,\,\,\,\,2\left| {\,\,\,-524} \right.}}\\\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,\,\,\,\times } \right.\end{array}\)

**Q.2 Find the square root of 1332.25 ?**

Sol :

\(\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\left| {36.5} \right.\\\overline{{\,\,\,\,\,\,3\left| {\,\,\,\underline{{13}}\underline{{32}}.\underline{{25}}} \right.}}\\\,\,\,\,\,\,3\left| {\,\,\,\,\,9} \right.\\\overline{{\,\,\,66\left| {\,\,\,\,\,432\,\,\,\,\,\,\,\,} \right.}}\\\underline{{\,\,\,\,\,\,6\left| {\,-396\,\,\,\,\,\,\,\,} \right.}}\\\,725\left| {\,\,\,\,\,\,\,\,3625} \right.\\\underline{{\,\,\,\,\,\,5\left| {\,\,\,\,\,-3625} \right.}}\\\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times } \right.\end{array}\)

**Q.3 The students of Class VIII of a school donated $ 2401 in all, for prime minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.**

Sol : Let’s the No. of students are = x

Each students donated = x

According to the questions,

x × x = 2401

x^{2} = 2401

x = √2401

\(\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\left| {49} \right.\\\overline{{\,\,\,\,\,\,4\left| {\,\,\,\underline{{24}}\underline{{01}}} \right.}}\\\,\,\,\,\,\,4\left| {-16} \right.\\\overline{{\,\,\,89\left| {\,\,\,\,\,\,801\,\,} \right.}}\\\underline{{\,\,\,\,\,\,9\left| {\,-801\,\,\,\,\,\,\,\,} \right.}}\\\,\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,\,\,\times } \right.\end{array}\)

So, total no. of students are 49 and each students gives $ 49.

**Q.4 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.**

Sol : Let’s the no of rows is = x

and the no of plants is = x

according to the question,

x × x = 2025

x^{2} = 2025

x = √2025

\(\displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\left| {45} \right.\\\overline{{\,\,\,\,\,\,4\left| {\,\,\,\underline{{20}}\underline{{25}}} \right.}}\\\,\,\,\,\,\,4\left| {-16} \right.\\\overline{{\,\,\,\,85\left| {\,\,\,\,\,425\,} \right.}}\\\underline{{\,\,\,\,\,\,5\left| {\,\,-425\,} \right.}}\\\,\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,\,\,\times } \right.\end{array}\)

So, the number of rows and the number of plants in each row are 45.

**Q.5 Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.**

Sol : First we do prime factors of 4, 9 and 10

\(\displaystyle \begin{array}{l}\underline{{\left. 2 \right|4,\,9,\,10}}\\\underline{{\left. 2 \right|2,\,9,\,5}}\\\underline{{\left. 3 \right|1,\,9,\,5}}\\\underline{{\left. 3 \right|1,\,3,\,5}}\\\underline{{\left. 5 \right|1,\,1,\,5}}\\\left. {\,\,} \right|1,\,1,\,1\end{array}\)

Prime factors are = 2 × 2 × 3 × 3 × 5

Here we can see that 5 group are not formed it mean this is not a perfect square number so we multiply it with 5

then 2 × 2 × 3 × 3 × 5 × 5

= 900

the smallest square number that is divisible by each of the numbers 4, 9 and 10 is 900.

**Q.6 Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.**

Sol : First we do prime factors of 8, 15 and 20

\(\displaystyle \begin{array}{l}\underline{{\left. 2 \right|8,\,15,\,20}}\\\underline{{\left. 2 \right|4,\,15,\,10}}\\\underline{{\left. 2 \right|2,\,15,\,5}}\\\underline{{\left. 3 \right|1,\,15,\,5}}\\\underline{{\left. 5 \right|1,\,5,\,5}}\\\left. {\,\,} \right|1,\,1,\,1\end{array}\)

Prime factors are = 2 × 2 × 2 × 3 × 5

Here we can see that 2, 3 and 5 group are not formed it mean this is not a perfect square number so we multiply it with 2, 3 and 5.

then, 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

= 3600

the smallest square number that is divisible by each of the numbers 8, 15 and 20 is 3600.

**Q.7 Find the least number that must be substracted from 5607 so as to get a prefect square. Also find the square root of the perfect square.**

Sol : Let us try to find √5607 by long division method.

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,74\\\,\,\,\,\,\overline{{7\left| {\,\,\,\,5607} \right.}}\\\left. {\,\,\,\,\,7} \right|-49\\\overline{{144\left| {\,\,\,\,\,\,707} \right.}}\\\,\,\,\,\,4\left| {\,\,-576} \right.\\\overline{{\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,131} \right.}}\end{array}\)

We get the remainder 131. It shows that 74^{2} is less than 5607 by 131.

this means if we subtract the remainder from the number, we get a perfect square.

Therefore, the required perfect square is 5607 – 131 = 5476. And √5476 = 74.

**Q.8 Find the greatest 4 digit number which is a perfect square.**

Sol : Greatest number of 4-digit = 9999.

Now we find √9999 by long division method.

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,99\\\,\,\,\,\,\overline{{9\left| {\,\,\,\,9999} \right.}}\\\left. {\,\,\,\,\,9} \right|-81\\\overline{{189\left| {\,\,\,\,1899} \right.}}\\\,\,\,\,\,9\left| {-1701} \right.\\\overline{{\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,198} \right.}}\end{array}\)

The remainder is 198. This shows 99^{2} is less than 9999 by 198.

This means if we subtract the remainder from the number, we get a perfect square.

Therefore, the required perfect square is 9999 – 198 = 9801.

And √901 = 99

**Q.9 Find the least number that must be added to 1300 so as to get a perfect square. Also find the square root of the perfect square.**

Sol : We find √1300 by long division method.

\(\displaystyle \begin{array}{l}\left. {\,\,\,\,\,\,\,\,} \right|\,\,\,\,\,36\\\,\,\,\,\,\overline{{3\left| {\,\,\,\,1300} \right.}}\\\left. {\,\,\,\,\,3} \right|\,\,-9\\\overline{{\,\,66\left| {\,\,\,\,\,\,\,400} \right.}}\\\,\,\,\,6\left| {\,\,\,\,-396} \right.\\\overline{{\,\,\,\,\,\,\,\,\left| {\,\,\,\,\,\,\,\,\,\,\,\,4} \right.}}\end{array}\)

The remainder is 4.

This shows that 36^{2} < 1300.

Next perfect square number is 37^{2} = 1369.

Hence, the number to be added is 37^{2} = 1300 = 1369 – 1300 = 69.