# If X = √ 6 + √ 5 ,Then X 2 + 1 X 2 − 2 = - Mathematics

MCQ

If $x = \sqrt{6} + \sqrt{5}$,then $x^2 + \frac{1}{x^2} - 2 =$

#### Options

• $2\sqrt{6}$

• $2\sqrt{5}$

• 24

• 20

#### Solution

Given that x = sqrt6 +sqrt5  .Hence 1/xis given as

1/x = 1/(sqrt6+sqrt5).We need to find  x^2 +1/x^2 - 2

We know that rationalization factor for  sqrt6 +sqrt5 issqrt6 -sqrt5. We will multiply numerator and denominator of the given expression   1/(sqrt6+sqrt5)by sqrt6 -sqrt5, to get

1/x = 1/(sqrt6+sqrt5) xx (sqrt6-sqrt5)/(sqrt6-sqrt5)

 = (sqrt6-sqrt5)/((sqrt6)^2 - (sqrt5)^2)

 = (sqrt6  - sqrt5)/(6-5)

 = sqrt6 - sqrt5.

We know that (x-1/x)^2 = x^2 + 1/x^2 - 2  therefore,

x^2 + 1/x^2 - 2 = (x-1/x)^2

 = (sqrt 6 + sqrt5 - (sqrt6  - sqrt5))^2

 = (sqrt6 + sqrt5 - sqrt6 +sqrt5)^2

 = (2sqrt5)^2

= 20

Concept: Laws of Exponents for Real Numbers
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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 3 Rationalisation
Exercise 3.4 | Q 24 | Page 18

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