## Square Root and Cube Root

### Important Formulae

1. Square Root:

If x^2 = y, we say that the square root of y is x and we write √y = x.

Thus, √4 = 2, √9 = 3, √196 = 14.

2. Cube Root:

The cube root of a given number x is the number whose cube is x.

We, denote the cube root of x by 3√x

Thus, √8 = 2 x 2 x 2 = 2, √343 = 7 x 7 x 7 = 7

Note:

1. √xy = √x x √y

2.√(x/y) = √x/√y =(√x/√y)*(√y/√y) = √(xy)/y

### General Tips

Example 1:

Find out the cube root of 50653?

Solution

The first step is to divide the number into 2 parts by separating the last 3 digits. So, we get 50 & 563 as the two parts of the number.

Now, take the first part and find the largest cube contained in the first part i.e. in 50 = 27 (which is the cube of 3). The next cube i.e. 64 (cube of 4) is larger than 50. Now, as 27 is the cube of 3, your ten’s part of cube root would be 3.

The next step is to take the last digit of the number, which in this case is 3. (7*7*7=343) Hence, 7 is the unit digit of your solution.

So, your answer is 37

Example 2:

Calculate the cube root of 941192?

By Step 1 —– We get two parts i.e. 941 and 192

By Step 2 —– The largest cube less than 941 is 729 (cube of 9). So, ten’s digit is 9.

By Step 3 —– The ending digit is 2. Hence, unit’s digit is 8. That’s it. 98 is the answer

Example 3:

The least perfect square, which is divisible by each of 21, 36 and 66 is:

Solution

L.C.M. of 21, 36, 66 = 2772.

Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11

To make it a perfect square, it must be multiplied by 7 x 11.

So, required number = 2^2 x 3^2 x 7^2 x 11^2 = 213444

Example 4:

√0.0169 x ? = 1.3

Solution:

Let √0.0169 x x = 1.3

Then, 0.0169x = (1.3)^2 = 1.69

x = 1.69/0.0169 = 100

Example 5:

If a = 0.1039, then the value of √4a2 - 4a + 1 + 3a is:

Solution:
√4a2 - 4a + 1 + 3a = √(1)2 + (2a)2 - 2 x 1 x 2a + 3a

= √(1 - 2a)2 + 3a

= (1 - 2a) + 3a

= (1 + a)

= (1 + 0.1039)

= 1.1039