Question

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

• A. 167
• B. 169
• C. 140
• D. 160

Answer 1

Answer: (C)

140

Given, $x-\frac{1}{x}=5$
Squaring both the sides,

$\Rightarrow (x-\frac{1}{x}{)}^2=5^2$

$\Rightarrow x^2+\frac{1}{x^2}-2\left(x\right)\left(\frac{1}{x}\right)=25$

$\Rightarrow x^2+\frac{1}{x^2}=25+2$

$\Rightarrow x^2+\frac{1}{x^2}=27$

Now,

$x^3-\frac{1}{x^3}$

$=(x-\frac{1}{x})\left[\right(x{)}^2+{\left(\frac{1}{x}\right)}^2+\left(x\right)\left(\frac{1}{x}\right)]$

$=5(27+1)$

$=140$