Question

If $a+\frac{1}{a}=2$, find $\frac{a^8+1}{a^4}$.

• A. $1$
• B. $7$
• C. $2$
• D. $5$

Answer 1

Answer: (C)

$2$

Given,$\frac{a^8+1}{a^4}$
$=\frac{a^8}{a^4}+\frac{1}{a^4}$
$=a^4+\frac{1}{a^4}$
Now,
$a+\frac{1}{a}=2$
Squaringbothsides,
$=>(a+\frac{1}{a}{)}^2=2^2$
$=>a^2+\frac{1}{a^2}+2\left(a\right)\left(\frac{1}{a}\right)=4$
$=>a^2+\frac{1}{a^2}=4-2$
$=>a^2+\frac{1}{a^2}=2$
Squaring both sides,we get.
$=>(a^2+\frac{1}{a^2}{)}^2=2^2$
$=>(a^2{)}^2+(\frac{1}{a^2}{)}^2+2\left(a^2\right)\left(\frac{1}{a^2}\right)=4$
$=>a^4+\frac{1}{a^4}=4-2$
$=>a^4+\frac{1}{a^4}=2$