Question

If $\left(\frac{a}{b}\right)+\left(\frac{b}{a}\right)=2$, then find ${\left(\frac{a}{b}\right)}^{10}-{\left(\frac{b}{a}\right)}^{10}$.

• A. $\frac{2^{20}-1}{2^{10}}$
• B. 2
• C. 0
• D. $\frac{2^{20}+1}{2^{10}}$

Answer 1

Answer: (C)

$\frac{a}{b}+\frac{b}{a}=2$
$\frac{a}{b}=t$
$\Rightarrow t+\frac{1}{t}=2$
$\Rightarrow t^2+1=2t$
$\Rightarrow t^2-2t+1=0$
$\Rightarrow (t-1{)}^2=0$
$t=1$
$t=\frac{a}{b}=1$
$\frac{1}{t}=\frac{b}{a}=1$
$(\frac{a}{b}{)}^{10}-(\frac{b}{a}{)}^{10}$
$\Rightarrow (1{)}^{10}-(1{)}^{10}$
$\Rightarrow 1-1$
$=0$