Given $x - \frac{1}{x} = 4$, find the value of $x^3 - \frac{1}{x^3}$.

1. Problem Description

Given $x - \frac{1}{x} = 4$, find the value of $x^3 - \frac{1}{x^3}$.

2. Solution Steps

We are given that $x - \frac{1}{x} = 4$. We need to find the value of $x^3 - \frac{1}{x^3}$. We can use the identity:

$a^3 - b^3 = (a-b)^3 + 3ab(a-b)$

Let $a = x$ and $b = \frac{1}{x}$. Then, we have:

$x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x)(\frac{1}{x})(x - \frac{1}{x})$

$x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(1)(x - \frac{1}{x})$

$x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x - \frac{1}{x})$

We know that $x - \frac{1}{x} = 4$. So, we can substitute this value into the equation:

$x^3 - \frac{1}{x^3} = (4)^3 + 3(4)$

$x^3 - \frac{1}{x^3} = 64 + 12$

$x^3 - \frac{1}{x^3} = 76$

3. Final Answer

The value of $x^3 - \frac{1}{x^3}$ is 76.