#### CUBE ROOT OF A NUMBER

**REMEMBERING UNITS DIGITS:**

First we need to remember cubes of 1 to 10 and unit digits of these cubes. The figure below shows the unit digits of cubes (on the right) of numbers from 1 to 10 (on the left).

Now look at the image above. We can definitely say that:

Whenever unit digit of a number is 9, the unit digit of the cube of that number will also be 9. Similarly, if the unit digit of a number is 9, the unit digit of the cube root of that number will also be 9. Similarly, if unit digit of a number is 2, unit digit of the cube of that number will be 8 and vice versa if unit digit of a number is 8, unit digit of the cube root of that number will be 2. Similarly, it will be applied to unit digits of other numbers as well.

**#2. DERIVING CUBE ROOT FROM REMAINING DIGITS**

Let’s see this with the help of an example. **Note that this method works only if the number given is a perfect cube.**

**Q. Find the cube root of 474552.**

Unit digit of 474552 is 2. So we can say that unit digit of its cube root will be 8.

Now we find cube root of 474552 by deriving from remaining digits.

Let us consider the remaining digits leaving the last 3 digits. i.e. 474.

Since 474 comes in between cubes of 7 and 8.

So the ten’s digit of the cube root will definitely be 7

i.e. cube root of 474552 will be 78.

** **

Let us take another example.

**Q. Find the cube root of 250047.**

Since the unit digit of the number is 7, so unit digit in the cube root will be 3.

Now we will consider 250.

Since, 6^{3} < 250 < 7^{3}, So tens digit will be 6

So we find cube root of the number to be 63.

Here are some more examples.

**Q. Simplify: ****∛****970299 = ?**

Unit digit of number is 9

∴ Unit digit of cube root will be 9

Now we will consider 970

Since, 9^{3} < 970 < 10^{3}

So, we find cube root of the number to be 99.

**Let’s take another example to make this trick clearer to you.**

**Q. Simplify: ****∛****140608**** = ?**

Unit digit of number is 8

∴ Unit digit of cube root will be 2

Now we will consider 140

Since, 5^{3} < 140 < 6^{3}

So, we find cube root of the number to be 52.

#### SQUARE ROOT OF A NUMBER

**Unit Digits of Squares**

First we need to remember unit digits of all squares from 1 to 10. The figure below shows the unit digits of the squares.

Now from the picture we can say that, **whenever the unit digit of a number is 9, unit digit of the square root of that number will be definitely 3 or 7**. Similarly, this can be applied to other numbers with different unit digits.**How to Find Square Root of a Four Digit Number**

Let’s learn how to find square root by taking different examples.

**Example:** Find the square root of 4489.

We group the last pair of digits, and the rest of the digits together.

Now, since the unit digit of 4489 is 9. So we can say that unit digit of its square root will be either 3 or 7.

Now consider first two digits i.e. 44. Since 44 comes in between the squares of 6 and 7 (i.e. 6^{2} < 44 < 7^{2}), so we can definitely say that the ten’s digit of the square root of 4489 will be 6. So far, we can say that the square root will be either 63 or 67.

Now we will find the exact unit digit.

To find the exact unit digit, we consider the ten’s digit i.e. 6 and the next term i.e. 7.

Multiply these two terms

Since, 44 is greater than 42. So square root of 4489 will be the bigger of the two options i.e. 67.

Let us take another example.

**Example: **What is the square root of 7056?

Unit digit will be 4 or 6.

Since, 8^{2} < 70 < 9^{2}

So the square root will be either 84 or 86.

Now consider 8 and 9

Since, 70 is less than 72. So square root will be the lesser of the two values i.e. 84.

Let’s try it out with five digit numbers now!

**How to Find Square Root of a Five Digit Number**

We pair the digits up starting from the right side. Since there is one extra left over after two pairs are formed, we club it with the pair closest to it.

**Example: **√(16641) = ?

Unit digit will be 1 or 9.

Since, 12^{2} < 166 < 13^{2}

So, the square root will definitely be 121 or 129.

Now, consider 12 and 13

Since, 166 is greater the 156, we pick the larger of the options i.e. 129.

Let’s take another example, so that this trick will be clear to you.

**Example: **√(33489) = ?

Unit digit will be 3 or 7.

Since, 18^{2} < 334 < 19^{2}

So, square root will be 183 or 187.

Now consider 18 and 19.

Now, 334 is less than 342. So, the square root will be lesser of the two numbers i.e. 183.

#### The cube root of .000216 is:

#### What should come in place of both x in the equation x / √128=√162 /x

#### (1.5625)^(1/2) = ?

#### The square root of (7 + 3√5) (7 - 3√5) is

#### (√625 x 14 x 11 / 11 x √25 x √196

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