# If X = 2 + Sqrt3,Find the Value of X^2 + 1/X^2 - Mathematics

if x = 2 +  sqrt3,find the value of x^2 + 1/x^2

#### Solution

We know that x^3 + 1/x^3 = (x + 1/x)(x^2 - 1 + 1/x^2). We have to find the value of x^3 + 1/x^3

As x = 2 + sqrt3 therefore

1/x = 1/(2 + sqrt3)

We know that rationalization factor for 2 + sqrt3 is 2 - sqrt3. We will multiply numerator and denominator  of the given expression 1/2 + sqrt3 by 2 - sqrt3 to get

1/x = 1/(2 + sqrt3) xx (2 - sqrt3)/(2 - sqrt3)

= (2 - sqrt3)/((2)^2 - (sqrt3)^2)

= (2 - sqrt3)/(4 - 3)

= 2 - sqrt3

Putting the value of x and 1/x we get

x^3 + 1/x^3= (2 + sqrt3 + 2 - sqrt3)((2 + sqrt3)^2 -  1+ (2 - sqrt3)^2)

= 4(2^2  + (sqrt3))^2 + 2 xx 2 xx sqrt3 - 1 + 2^2  + (sqrt3)^2 - 2 xx 2 xx sqrt3)

= 4(4 + 3 + 4sqrt3 - 1 + 4 + 3 - 4sqrt3)

= 52

Hence the value of the given expression 52.

Concept: Operations on Real Numbers
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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 3 Rationalisation
Exercise 3.2 | Q 10 | Page 15

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