Question

Find the smallest number by which 2400 is to be multiplied to get a perfect square and also find the square root of the resulting number.

Answer 1

Hint: Since we just need to know such things about the square root numbers and perfect square numbers, An perfect square is the numbers that obtain by multiplying any whole numbers (zero to infinity) twice, or the square of the given numbers yields a whole number like $\sqrt {25} = 5$ or $25 = {5^2}$ needs to satisfy both and square root are the terms like $\sqrt 9 = 3$ (times of the common numbers)

Complete step-by-step solution:
Let we need to find the number which multiplied $2400$ to get the perfect square, so first, rewrite the given number into the form of square root separation and then we can find the perfect square; $2400 = 6 \times 4 \times 100$ after separating now we need to eliminate the non-perfect square numbers. since $4$ is a perfect square by the perfect square definition $4 = {2^2}$ or $\sqrt 4 = 2$ hence eliminate the numbers $4$ also $100 = {10^2}$ or $\sqrt {100} = 10$ hence hundred is also the perfect square and will be eliminated. Thus the only number remaining is $6$ which is not the perfect square number, hence the smallest number is $6$ by which $2400$ is to be multiplied to get a perfect square, thus we get $2400 \times 6 = {6^2} \times {2^2} \times {10^2}$ hence the answer is $2400 \times 6 = 14400$ which is a perfect square because $14400 = {120^2}$
Hence also the square root of the resulting number is $\sqrt {14400} = 120$ (by taking square root)
Thus, we get $6$ is the number that multiplied to get perfect square for $2400$ as $14400$ also the square root of the number is $120$ this the result

Note: Perfect square and square root has only difference which is perfect needs to satisfy both common terms in square and square root while the square root is all about only common terms like $\sqrt 9 = 3$