If `x+\frac{1}{x}=3`, find `x^3+\frac{1}{x^3}`?

Mathematics often presents us with intriguing problems that require a blend of algebraic manipulation and logical reasoning.  Try to solve the following problem.

If `x+\frac{1}{x}=3`, find `x^3+\frac{1}{x^3}`?

This is very easy one. Even Grade 8 students also can do it.

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Answer: 18

Explanation: 

`\left( x+\frac{1}{x} \right)^3=x^3+3x+\frac{3}{x}+\frac{1}{x^3}`

`\left( x+\frac{1}{x} \right)^3=x^3+3\left(x+\frac{1}{x}\right)+\frac{1}{x^3}`

`\left(3 \right)^3=x^3+3\left(3\right)+\frac{1}{x^3}`

`27=x^3+9+\frac{1}{x^3}`

`x^3+\frac{1}{x^3}=27-9=18`