Question

Evaluate this please​

Answer 1

ANSWER:

To Evaluate:

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

Solution:

We are given that,

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

We know that,

$\hookrightarrow a^{m-n}=\dfrac{a^m}{a^n}$

So,

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

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

On taking LCM,

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

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

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

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

$\implies\dfrac{x^p}{x^p+x^q}+\dfrac{x^q}{x^p+x^q}$

So,

$\implies\dfrac{x^p+x^q}{x^p+x^q}$

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

$\implies 1$

Therefore,

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