Question

Find the value of $\dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}}$ is equal to
A) $\dfrac{{{x^{ab}}}}{{{x^a} + {x^b}}}$
B) $\dfrac{{{x^{ab}}}}{{{x^{a - b}}}}$
C) $1$
D) $\dfrac{{{x^{ab}}}}{{{x^{b - a}}}}$

Answer 1

Hint: First use the law of indices which is ${x^{m - n}} = \dfrac{{{x^m}}}{{{x^n}}}$. After that take LCM in the denominator. Then use the identity $\dfrac{1}{{\dfrac{a}{b}}} = \dfrac{b}{a}$ to simplify the expression. After that again take LCM and add the terms in the denominator. Then cancel out the common factors to get the desired result.

Complete step-by-step solution:
Simplify $\dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}}$ by using the law of indices.
Use the law of indices \[{x^{m - n}} = \dfrac{{{x^m}}}{{{x^n}}}\] and rewrite the terms,
$ \Rightarrow \dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = \dfrac{1}{{1 + \dfrac{{{x^a}}}{{{x^b}}}}} + \dfrac{1}{{1 + \dfrac{{{x^b}}}{{{x^a}}}}}$
Take LCM in the denominator of both fractions,
$ \Rightarrow \dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = \dfrac{1}{{\dfrac{{{x^b} + {x^a}}}{{{x^b}}}}} + \dfrac{1}{{\dfrac{{{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.
$ \Rightarrow \dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = \dfrac{{{x^b}}}{{{x^b} + {x^a}}} + \dfrac{{{x^a}}}{{{x^a} + {x^b}}}$
Again, take LCM of both the fractions,
$ \Rightarrow \dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = \dfrac{{{x^b} + {x^a}}}{{{x^b} + {x^a}}}$
Cancel out the common factors from numerator and denominator,
$\therefore \dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = 1$
Thus, the value of $\dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}}$ is 1.

Hence, option (C) is the correct answer.

Note: When the base is the same then only we can apply the law of indices, be it the multiplication of numbers with the same base or division of numbers with the same base.
Always keep in mind that any number with power 0 will be equal to 1.
${a^\circ} = 1$
If powers are in fraction form, then we can write them as
${\left( a \right)^{\dfrac{m}{n}}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}}$
which means we take ${n^{th}}$ root of the term inside the bracket, so we can write
${\left( a \right)^{\dfrac{m}{n}}} = \sqrt[n]{{{a^m}}}$
Any number with the power m means that the number is multiplied to itself m number of times. So, we can write
${x^m} = \underbrace {x \times x \times x \times \ldots \ldots \times x}_m$