Question

If $x^4+\frac{1}{x^4}=322$, then find the value of $(x^3-\frac{1}{x^3})$.

• A. $109$
• B. $105$
• C. $100$
• D. $76$

Answer 1

The correct option is D $76$

$x^4+\frac{1}{x^4}=322$

$x^4+\frac{1}{x^4}+2=324$

${(x^2+\frac{1}{x^2})}^2={18}^2$

$(x^2+\frac{1}{x^2})=18$ ............................(I)

$x^2+\frac{1}{x^2}-2=16$

${(x-\frac{1}{x})}^2=16$

$x-\frac{1}{x}=4$ ...........................(II)

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

$=4\times (18+1)$

$=76$