Question

What is the smallest number by which √5–√2 is to be multiplied to make it a rational number? Also find the number so obtained?

Answer 1

To make rational we have to multiply with its conjugate.
Here conjugate of (√5 - √2) is (√5 +√2)
(√5 - √2) (√5 +√2)
(√5)² - (√2)²

[(a+b)(a-b)= a² -b²]
= 5 - 2
= 3
3 is a rational number.
Hence,the smallest number by which √5–√2 is to be multiplied to make it a rational number is √5+√2 and the number so obtained is 3.

HOPE THIS WILL HELP YOU...

Answer 2

$\sqrt{5 \:} - \sqrt{2}$
CONJUGATE OF THIS IS : -
$\sqrt{5 \:} \: + \sqrt{2}$
IN ORDER , TO MAKE A NUMBER RATIONAL WE NEED TO MULTIPLE THE NO. WITH IT'S CONJUGATE !

I.E. = $\sqrt{5} \: - \sqrt{2} \: \times \sqrt{5} + \sqrt{2}$
USING , ${(a - b)}^{2} = (a - b)(a + b)$

WE GET ,

= 5 - 2
= 3

Therefore, The Number so obtained is 3 !

Hope this helps !