Question

Prove that $\dfrac{a+b+c}{{{a}^{-1}}{{b}^{-1}}+{{b}^{-1}}{{c}^{-1}}+{{c}^{-1}}{{a}^{-1}}}=abc$ .

Answer 1

Hint: In the given question we are given an equality which we need to prove that the left-hand side is equal to the right-hand side of the given equality. Also, we need to make sure that we have the common denominator for all the terms so that we can prove the given equality easily.

Complete step-by-step solution:
In the given question we are given that, $\dfrac{a+b+c}{{{a}^{-1}}{{b}^{-1}}+{{b}^{-1}}{{c}^{-1}}+{{c}^{-1}}{{a}^{-1}}}=abc$
Now, we need to show that both sides of the given equation are equal. So, for this we will proceed with the left-hand side of the equation.
Now, taking the left-hand side we have $\dfrac{a+b+c}{{{a}^{-1}}{{b}^{-1}}+{{b}^{-1}}{{c}^{-1}}+{{c}^{-1}}{{a}^{-1}}}$
Now, simplifying this we get,
$\begin{align}
  & \dfrac{a+b+c}{{{a}^{-1}}{{b}^{-1}}+{{b}^{-1}}{{c}^{-1}}+{{c}^{-1}}{{a}^{-1}}} \\
 & \Rightarrow \dfrac{a+b+c}{\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac}} \\
\end{align}$
Now, for the denominator term of the given fraction what we can do is multiply and divide first, second and third term by c, a, b respectively and we will get,
$\begin{align}
  & \dfrac{a+b+c}{\dfrac{c}{acb}+\dfrac{a}{abc}+\dfrac{b}{abc}} \\
 & \Rightarrow \dfrac{\left( a+b+c \right)abc}{a+b+c} \\
 & \Rightarrow abc \\
\end{align}$
Now, we can see that as soon as the denominator becomes the same we get the same terms in the numerator and denominator and cancelling them gives us abc which is the same as the right-hand side of the equation.

Note: The given question is easy to prove and we can prove right-hand side equal to left-hand side but we can clearly see that proceeding from right-hand side will lead to complications, so in order to avoid that we will proceed with left-hand side and try to reduce the calculation mistakes rest th question is easy to attempt.