Find the Value
\[ x-\frac{1}{x}=7 \]
Find:
\[ x^3-\frac{1}{x^3} \]
Solution:
Using identity:
\[ a^3-b^3=(a-b)^3+3ab(a-b) \]
Here,
\[ a=x,\quad b=\frac{1}{x} \]
\[ ab=x\left(\frac{1}{x}\right)=1 \]
\[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x\cdot\frac{1}{x}\right)\left(x-\frac{1}{x}\right) \]
\[ = (7)^3+3(1)(7) \]
\[ = 343+21 \]
\[ =364 \]