Question

if x =root 3 + root 2 find value of x cube -1/x cube​

Answer 1

Answer :

x³ - 1/x³ = 22√2

Solution :

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

We have ,

x = √3 + √2

Thus ,

1/x = 1/(√3 + √2)

Now ,

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

=> 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 = √3 + √2 - √3 + √2

=> x - 1/x = 2√2

Now ,

We know that ,

(A - B)³ = A³ - B³ - 3AB(A - B)

If A = x and B = 1/x , then

=> (x - 1/x)³ = x³ - (1/x)³ - 3•x•(1/x)•(x - 1/x)

=> (x - 1/x)³ = x³ - 1/x³ - 3•(x - 1/x)

=> (2√2)³ = x³ - 1/x³ - 3•2√2

=> 16√2 = x³ - 1/x³ - 6√2

=> x³ - 1/x³ = 16√2 + 6√2

=> x³ - 1/x³ = 22√2

Hence ,

x³ - 1/x³ = 22√2