Question

if x=√3+√2 , find the value of x+1/x+x^3+1/x^3​

Answer 1

Answer :

x + 1/x + x³ + 1/x³ = 20√3

Solution :

• Given : x = √3 + √2
• To find : x + 1/x + x³ + 1/x³

We have ,

x = √3 + √2

Thus ,

1/x = 1/(√3 + √2)

Now ,

Rationalising the denominator of the term in RHS , we have ;

=> 1/x = (√3 - √2)/(√3 + √2)(√3 - √2)

=> 1/x = (√3 - √2)/[(√3)² - (√2)²]

=> 1/x = (√3 - √2)/(3 - 2)

=> 1/x = (√3 - √2)/1

=> 1/x = √3 - √2

Now ,

=> x + 1/x = (√3 + √2) + (√3 - √2)

=> x + 1/x = 2√3

Now ,

Cubing both the sides , we get ;

=> (x + 1/x)³ = (2√3)³

=> x³ + (1/x)³ + 3•x•(1/x)•(x + 1/x) = 24√3

=> x³ + 1/x³ + 3•1•2√3 = 24√3

=> x³ + 1/x³ + 6√3 = 24√3

=> x³ + 1/x³ = 24√3 - 6√3

=> x³ + 1/x³ = 18√3

Now ,

=> x + 1/x + x³ + 1/x³ = 2√3 + 18√3

=> x + 1/x + x³ + 1/x³ = 20√3

Hence ,

x + 1/x + x³ + 1/x³ = 20√3