Question

Prove that $\dfrac{1}{{1 + {x^{a - b}}}} + \dfrac{1}{{1 + {x^{b - a}}}} = 1$

Answer 1

Hint: If the student gives a closer look to the sum, they will understand that it is just a single step sum. It involves application of the identity \[\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}\].
If the student is aware of this identity then he just has to apply it and then proceed with simple addition of fractions. Whenever a student gets a sum wherein he has to prove LHS is equal to RHS , it is advisable to solve LHS & RHS partially and prove it , rather than Reducing LHS to the form of RHS. This is because in the second case the numerical would become much more complicated. In this particular sum , since the RHS is a constant we will have to solve LHS and prove that it is equal to $1$.

Complete step-by-step answer:
It involves usage of the Identity \[\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}\].
Thus we can bring the given question to the form of the identity to make sum look much easier.
Let us consider the LHS only and solve the problem .
$\Rightarrow$ $\dfrac{1}{{1 + \dfrac{{{x^a}}}{{{x^b}}}}} + \dfrac{1}{{1 + \dfrac{{{x^b}}}{{{x^a}}}}}.................(1)$
Now,taking the LCM of the denominator for both the fractions separately we get the following step
$\Rightarrow$ $\dfrac{{{x^b}}}{{{x^b} + {x^a}}} + \dfrac{{{x^a}}}{{{x^a} + {x^b}}}.................(2)$
From this step, the student can gauge that he has solved the sum and is just a single step away. Since both the fractions have a common denominator we can combine them and add the fractions simply.
$\Rightarrow$ $\dfrac{{{x^b} + {x^a}}}{{{x^b} + {x^a}}}.....................(3)$
Since the numerator and denominator are the same, we can strike it off. Thus the value of this fraction is $1$.
LHS =RHS

Hence Proved.

Note: In order to solve such a numerical ,knowledge about indices and its properties would be beneficial. Sometimes the problems on the chapter of indices are purely based on the properties of Indices. For example the student may be given a sum ${3^2}$ $ \times $ ${3^{\dfrac{2}{3}}}$ $ \times $ ${2^2}$$ \times $ ${2^{-2}}$ and have to find its value. In this sum the student would have to apply the properties of Indices instead of calculating it manually. Thus a student should always learn the basic properties of Indices.