Question

If x = 3+2√2 , then find the value of x³+1/x³

Answer 1

Solution :-

Given : x = 3 + 2√2

To find : x³ + 1/x³

1/x = 1/(3 + 2√2)

On rationalizing :

= 1/(3 + 2√2) × (3 - 2√2) /(3 - 2√2)

= 3 - 2√2 / 9 - 8

= 3 - 2√2

x³ = (3 + 2√2)³

By using identity (a + b)³

= 3³ + (2√2)³ + 3 × 3 × 2√2 (3 + 2√2)

= 27 + 16√2 + (18√2 × 3) + (18√2 × 2√2)

= 27 + 16√2 + 54√2 + 72

= 99 + 70√2

1/x³ = (3 - 2√2)³

By using identity (a - b)³

= 3³ - (2√2)³ - 3 × 3 × 2√2 (3 - 2√2)

= 27 - 16√2 - (18√2 × 3) + (18√2 × 2√2)

= 27 - 16√2 - 54√2 + 72

= 99 - 70√2

Now,

x³ + 1/x³ = (99 + 70√2) + (99 - 70√2)

= 99 + 70√2 + 99 - 70√2

= 198

Hence,

198 is the required answer.