If:
X+1/X=4
then what is the value of
X^3+1/x^3
Hello, Rimas!
This problem has a spectacular solution . . .
If $\displaystyle x + \frac{1}{x} \,=\,4$
then what is the value of $\displaystyle x^3 + \frac{1}{x^3}$
Cube the equation: .$\displaystyle \left(x + \frac{1}{x}\right)^3\;=\;4^3$
We have: .$\displaystyle x^3 + 3x^2\!\cdot\!\frac{1}{x} + 3x\!\cdot\!\frac{1}{x^2} + \frac{1}{x^3} \:= \:64\quad\Rightarrow\quad x^3 + 3x + \frac{3}{x} + \frac{1}{x^3}\:=\:64$
Then: .$\displaystyle x^3 + 3\underbrace{\left(x + \frac{1}{x}\right)}_\downarrow + \frac{1}{x^3} \:=\:64$
. . . . . . . $\displaystyle x^2 + 3(4) + \frac{1}{x^3} \:=\:64$
. . . . . . . . . $\displaystyle x^3 + \frac{1}{x^3} \:=\:52$