Question

If $a^4+\frac{1}{a^4}=1154$, then the value of $a^3+\frac{1}{a^3}$ is

• A. $198$
• B. $216$
• C. $200$
• D. $196$

Answer 1

The correct option is A $198$

$a^4+\frac{1}{a^4}=1154$ $\Rightarrow$ $a^4+2+\frac{1}{a^4}=1154+2$
$\Rightarrow$${(a^2+\frac{1}{a^2})}^2=1156\Rightarrow a^2+\frac{1}{a^2}=34$
$\Rightarrow a^2+2+\frac{1}{a^2}=34+2\Rightarrow {(a^{}+\frac{1}{a^{}})}^2=36$
$\Rightarrow a^{}+\frac{1}{a^{}}=6$
$\therefore {(a^{}+\frac{1}{a^{}})}^3=216$
$\Rightarrow a^3+\frac{1}{a^3}+3a\times \frac{1}{a}(a^{}+\frac{1}{a^{}})=216$
$\therefore a^3+\frac{1}{a^3}+3\times 6=216$
$\Rightarrow a^3+\frac{1}{a^3}=216-18=198$