Question

If `a/b + b/a = 1` then the value of a³ + b³ is ______.

पर्याय

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Answer 1

If `a/b + b/a = 1` then the value of a³ + b³ is 0.

Explanation:

Given that `a/b + b/a = 1`,

We multiply both sides by (ab): a² + b² = ab.

We want to find the value of a³ + b³.

Recall the identity: a³ + b³ = (a + b)³ – 3ab(a + b)

First, calculate (a + b)²: (a + b)² = a² + 2ab + b² 

Using a² + b² = ab, 

Substitute: (a + b)² = ab + 2ab = 3ab 

Thus: `a + b = sqrt(3ab)`.

Now, substitute into the cube formula:

a³ + b³ = (a + b)³ – 3ab(a + b)

`a^3 + b^3 = (sqrt(3ab))^3 - 3ab sqrt(3ab)`

`a^3 + b^3 = 3sqrt(3)(ab)^(3//2) - 3sqrt(3)(ab)^(3//2)`

a³ + b³ = 0

So, the value of a³ + b³ is 0.