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

Polynomials, LCM and HCF of Polynomials

Solution:

$\frac{a}{b}+\frac{b}{a}=2$
$ \frac{a}{b}=t $
$\Rightarrow t+\frac{1}{t}=2$
$ \Rightarrow t^2+1=2 t$
$ \Rightarrow t^2-2 t+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$