Question

Find the smallest number by which 11250 must be multiplied in order to get a perfect square. Also find the square root of the new number.

Answer 1

Sᴏʟᴜᴛɪᴏɴ :-

According to the question , We have to find the smallest number by which 11250 should be multiplied so that the product is a perfect square.

For that we will first do prime factorisation of 11250 .

→ 11250 = 2 x 3 x 3 x 5 x 5 x 5 x 5.

→ Written in exponential form = 2¹ x 3² x 5⁴

we can see that 2 is left unpaired . To make 11250 a perfect square , 2 should be paired .

So, By this we can see that if we multiple 11250 by 2 we will get perfect square.

So,

11250 × 2 = 22500 (2² × 3² × 5⁴)

Hence , the smallest multiple of 11250 which is a perfect Square is = 22500 .

Now,

→ The square root of obtained perfect square = √22500 = √(2² × 3² × 5⁴) = 2 × 3 × 5 × 5 = 150 (Ans.)

Answer 2

QUESTION :-

Find the smallest number by which 11250 must be multiplied in order to get a perfect square. Also find the square root of the new number.

GIVEN :-

A number 11250, from which we need to divide the smallest number which gives a perfect square as a quotient.

TO FIND :-

The smallest that must be divided from 11250.

SOLUTION :-

Prime factorising 11250 :-

$\begin{array}{r | l}2 & 11250 \\\cline{2-2} 5 & 5625 \\\cline{2-2} 5 & 1125 \\\cline{2-2} 5 & 225 \\\cline{2-2} 5 & 45 \\\cline{2-2} 3 & 9 \\\cline{2-2} 3 & 3 \\\cline{2-2} & 1\end{array}$

11250 = 2 × 5 × 5 × 5 × 5 × 3 × 3

⇒11250 = 2 × 5² × 5² × 3²

We can see that, 2 is not in a pair. So, 2 is needed to be paired to make 11250 a perfect square.

11250 × 2 = 2 × 5² × 5² × 3² × 2

(2 is multiplied to both the sides)

⇒22500 = 2² × 5² × 5² × 3²

Here, 22500 is a perfect square.

So, the smallest number by which 11250 must be multiplied in order to get a perfect square is '2'.

$\rule{200}{2}$

Now,

Square root of 22500 = 15 × 15 × 10 × 10

⇒22500 = 15² × 10²

⇒√22500 = √15² × 10²

⇒√22500 = 15 × 10         (Squares get cancelled)

⇒√22500 = 150

Therefore, the square root of the new number is 150.