Question

Find the least rationalising factor of root 2 + root 7 - root 10

Answer 1

Rationalising factor means, the term by which we convert irrational number to rational number . so, it means I have to choose a term by which we make (√2 + √7 - √10) is a rational number.
Let's try to solve .
(√2 + √7 - √10)(√2 + √7 + √10) = (√2 + √7)² - √10²
= 2 + 7 + 2√14 - 10
= 2√14 - 1
= √56 - 1

again, (√56 - 1)(√56 + 1) = 56 - 1 = 55
e.g., (√2 + √7 - 10)[(√2 + √7 + √10)(√56 - 1)] = 55
So, rationalising factor is (√2 + √7 + √10)(√56 + 1)
You can resolve it ,
(√2 + √7 + √10)(√56 + 1)
= √112 + √392 + √560 + √2 + √7 + √10
= 4√7 + 14√2 + 4√35 + √2 + √7 + √10
= 5√7 + 15√2 + 4√35 + √10 [ ans ]

Answer 2

Rationalising factor is a term with which a term is multiplied or divided to make the whole term rational.

So, lets try to solve:-

Multiplying the given term with (√2 + √7 + √10)

(√2 + √7 - √10)(√2 + √7 + √10) = (√2 + √7)^2 - 10 = 2√14 - 1. 

and 
(√2 + √7 - √10)[(√2 + √7 + √10) (2√14 + 1)] 
= (2√14 - 1)(2√14 + 1) 
= 55. 

Thus, (√2 + √7 + √10) (2√14 + 1) 
= (2√28 + 2√98 + 2√140) + (√2 + √7 + √10) 
= (4√7 + 14√2 + 4√35) + (√2 + √7 + √10) 
= 15√2 + 5√7 + √10 + 4√35 is one such rationalizing factor.