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Find the value of $\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}$ is equal to

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\frac{x^{ab}}{(x^{a} + x^{b})}

\frac{x^{ab}}{x^{a - b}}

1

\frac{x^{ab}}{x^{b - a}}

answer is C.

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Detailed Solution

Simplify

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

by using the law of indices.
Use the law of indices

x^{m - n} = \frac{x^{m}}{x^{n}}

and rewrite the terms,
⇒

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

\frac{1}{1 + \frac{x^{a}}{x^{b}}} + \frac{1}{1 + \frac{x^{b}}{x^{a}}}

Take LCM in the denominator of both fractions,
⇒

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

\frac{\frac{1}{x^{b} + x^{a}}}{x^{b}} + \frac{\frac{1}{x^{a} + x^{b}}}{x^{a}}

Remember, that dividing by fraction is the same as flipping the fraction in the denominator and then multiplying it with the numerator.
So, flip the fraction in the denominator and multiply it to the numerator.
⇒

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

\frac{x^{b}}{x^{b} + x^{a}} + \frac{x^{a}}{x^{a} + x^{b}}

Again, take LCM of both the fractions,
⇒

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

\frac{x^{b} + x^{a}}{x^{b} + x^{a}}

Cancel out the common factors from numerator and denominator,
∴

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

=1
Thus, the value of

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}}

is 1.
Hence, option (3) is the correct answer.

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