Question

if x= 1+√2) , then the value of (x+1/x)^2 is​

Answer 1

Step-by-step explanation:

Given :-

x = 1+√2

To find :-

Find the value of (x+ 1/x)^2 ?

Solution:-

Given that

x = 1+√2

1/x = 1/(1+√2)

Denominator = 1+√2

We know that

The Rationalising factor of a+√b = a-√b

The Rationalising factor of 1+√2 is 1-√2

On Rationalising the denominator then

=> 1/x = [1/(1+√2)]×[(1-√2)/(1-√2)]

=> 1/x = (1-√2)/(1+√2)(1-√2)

The denominator is in the form of (a+b)(a-b)

Where a = 1 and b = √2

We know that

(a+b)(a-b)=a^2-b^2

=> 1/x = (1-√2)/(1^2-(√2)^2)

=> 1/x = (1-√2)/(1-2)

=>1/x = (1-√2)/(-1)

=> 1/x = -(1-√2)

=> 1/x = √2-1

Now

x + (1/x)

=> 1+√2 +√2-1

=> (1-1)+(√2+√2)

=> 0+(2√2)

=>2√2

Now the Value of (x+ 1/x)^2

=> (2√2)^2

=> 2^2×(√2)^2

=> 4×2

=> 8

Answer:-

The value of (x +1/x)^2 for the given problem is 8

Used formulae:-

Rationalising factor:-

The product of two irrational numbers is a rational number then they are called Rationalising factors of each other.

• The Rationalising factor of a+√b = a-√b
• (a+b)(a-b)=a^2-b^2
• (ab)^m = a^m × b^m