Question

Evaluate: x^p/x^p+x^q + 1/x^p-q+1.​

Answer 1

$\large\underline{\sf{Solution-}}$

We have to evaluate:

$\sf\dfrac{x^p}{x^p+x^q}+\dfrac{1}{x^{p-q}+1}$

We know that,

• $\sf a^{m-n}=\dfrac{a^m}{a^n}$

So,

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{1}{x^{p-q}+1}$

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{1}{\frac{x^p}{x^q}+1}$

On taking LCM,

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{1}{\frac{x^p}{x^q}+1}$

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{1}{\frac{x^p+x^q}{x^q}}$

Now, transposing $\sf x^q$ from denominator to numerator,

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{1}{\frac{x^p+x^q}{x^q}}$

$\sf\longmapsto\dfrac{x^p}{x^p+x^q}+\dfrac{x^q}{x^p+x^q}$

So,

$\sf\longmapsto\dfrac{x^p+x^q}{x^p+x^q}$

Cancelling $\sf x^p+x^q$ in both denominator and numerator,

$\sf\longmapsto 1$

Therefore,

$\boxed{\bf \dfrac{x^p}{x^p+x^q}+\dfrac{1}{x^{p-q}+1}=1}$