Question

Find the smallest natural number by which 1200 should be multiplied so that the square root of the product is a rational number.

Answer 1

Hint: In this particular type of question we have to proceed by taking the smallest natural number as x. Then we need to multiply it with 1200 and do the square root of the product to check the number which needs to be multiplied with 1200 to get the product a complete square.

Complete step-by-step answer:
Let the smallest natural number to be multiplied by x.
Multiplying and doing square root, we get
$\sqrt {1200 \times x} = \sqrt {2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times x} = 20\sqrt {3 \times x} $
Hence, 3 is the smallest natural number that needs to be multiplied with 1200, so as to make the product a rational number.

Note: Keep in mind that in this question we have to make the smallest complete square of the product of 1200 and x. Note that a number is a complete square when its root is a whole number. Also you should know that negative numbers are never a complete square.