Find the Value
\[ x+\frac{1}{x}=5 \]
Find:
\[ x^3+\frac{1}{x^3} \]
Solution:
Using identity:
\[ a^3+b^3=(a+b)^3-3ab(a+b) \]
Here,
\[ a=x,\quad b=\frac{1}{x} \]
\[ ab=x\left(\frac{1}{x}\right)=1 \]
\[ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3 -3\left(x\cdot\frac{1}{x}\right)\left(x+\frac{1}{x}\right) \]
\[ = (5)^3-3(1)(5) \]
\[ = 125-15 \]
\[ =110 \]