Question

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

• A. $76$
• B. $83$
• C. $71$
• D. $80$

Answer 1

The correct option is A $76$

$a^4+\frac{1}{a^4}=322\Rightarrow a^4+\frac{1}{a^4}+2=322+2$
$\Rightarrow {(a^2+\frac{1}{a^2})}^2=324$

$\Rightarrow \sqrt{324}=18$
$a^2+\frac{1}{a^2}-2=18-2$

$\Rightarrow {(a^{}+\frac{1}{a^{}})}^2=16$

$\Rightarrow a-\frac{1}{a}=\sqrt{16}=4$
$\therefore {(a-\frac{1}{a})}^3=a^3-\frac{1}{a^3}-3\times a\times \frac{1}{a}(a-\frac{1}{a})$
$\Rightarrow 4^3=a^3-\frac{1}{a^3}-3\times 4$

$\Rightarrow a^3-\frac{1}{a^3}=64+12=76$