If x 1 and x² + 1/x² = 83, then find the value of x³ - 1/x³.

Question

If x > 1 and x² + 1/x² = 83, then find the value of x³ - 1/x³.

Accepted Answer

Given: x > 1 and x² + 1/x² = 83.

Step 1: Let y = x + 1/x.

We know the identity:
(x + 1/x)² = x² + 2 + 1/x²
=> y² = x² + 1/x² + 2

Given x² + 1/x² = 83, so:

y² = 83 + 2 = 85

=> y = √85 (since x > 1, x + 1/x > 0, so positive root)

Step 2: Find x³ - 1/x³.

Recall the identity:
(x - 1/x)³ = x³ - 3x + 3(1/x) - 1/x³ = x³ - 1/x³ - 3(x - 1/x)

But this is complicated. Instead, use:

(x - 1/x)² = x² - 2 + 1/x² = (x² + 1/x²) - 2 = 83 - 2 = 81

So, x - 1/x = ±9.

Since x > 1, x - 1/x > 0, so x - 1/x = 9.

Step 3: Use the identity:
(x - 1/x)³ = x³ - 3x(1/x)(x - 1/x) - 1/x³ = x³ - 1/x³ - 3(x - 1/x)

Rearranged:
x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

Substitute x - 1/x = 9:

x³ - 1/x³ = 9³ + 3*9 = 729 + 27 = 756

Step 4: Check the options.

756 is option 3.

However, the options given are 876, 884, 756, 754.

So the correct answer is 756.

Answer

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