If $$x - \frac{1}{x} = 7,  then  x^3 - \frac{1}{x^3}$$ is equal to:

As we know the identity 

$$\therefore (a - b)^3 = a^3 - b^3 - 3ab(a - b)$$

Given,   $$x - \frac{1}{x} = 7$$

$$\Rightarrow (x - \frac{1}{x})^3 = 7^3$$

$$\Rightarrow x^3 - (\frac{1}{x})^3 - 3 (x)(\frac{1}{x})(x - \frac{1}{x}) = 343$$

$$\Rightarrow x^3 -\frac{1}{x^3}  - 3(x - \frac{1}{x}) = 343 $$

$$\Rightarrow x^3 -\frac{1}{x^3}  = 343 + 3(x - \frac{1}{x})$$

$$\Rightarrow x^3 -\frac{1}{x^3} = 343 + 3 \times 7 $$

$$\Rightarrow x^3 -\frac{1}{x^3} = 343 + 21 = 364 $$