Question

if x+ 1/x = 5, then find the value of x³+1/x³

Answer 1

GIVEN :

If $x+\frac{1}{x}=5$, then find the value of $x^3+\frac{1}{x^3}$

TO FIND :

The value of  $x^3+\frac{1}{x^3}$

SOLUTION :

Given that the value of $x+\frac{1}{x}=5$

Now solving $x+\frac{1}{x}=5$

Cubing on both sides,

$(x+\frac{1}{x})^3=5^3$

$(x +\frac{1}{x})^3=x^3+(\frac{1}{x})^3+3x^2(\frac{1}{x})^2+3x(\frac{1}{x})^2$

By using the Algebraic identity :

$( a + b )^3=a^3+ b^3 + 3a^2b+3ab^2$

Here the values are a = x and $b=\frac{1}{x}$

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

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

Substitute the value of $x+\frac{1}{x}=5$ in the above equation we have,

$(5)^3=x^3+(\frac{1}{x})^3+3(5)$

$125=x^3+\frac{1}{x^3}+15$

$125-15=x^3+\frac{1}{x^3}$

Rewritting the above equation we have that,

$x^3+\frac{1}{x^3}=125-15$

$x^3+\frac{1}{x^3}=110$

∴ the value of $x^3+\frac{1}{x^3}$ is 110.

∴ $x^3+\frac{1}{x^3}=110$