Question

Find the value of P if ${{x}^{a-b}}\times {{x}^{b-c}}\times {{x}^{c-a}}=P$ .
(a) 0
(b) 1
(c) ${{x}^{0}}$
(d) 10

Answer 1

Hint: We start solving the problem by recalling the law of exponents ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ and using it in the given ${{x}^{a-b}}\times {{x}^{b-c}}\times {{x}^{c-a}}=P$. We then make the necessary calculations to simplify the given equation. We then use another law of exponents ${{a}^{0}}=1$ to get the required answer(s) of the given answer.

Complete step by step answer:
According to the problem, we are given that ${{x}^{a-b}}\times {{x}^{b-c}}\times {{x}^{c-a}}=P$ and we need to find the value of P.
From the law of exponents, we know that ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$. We use this property in finding the value of P.
So, we get ${{x}^{a-b+b-c}}\times {{x}^{c-a}}=P$.
$\Rightarrow {{x}^{a-c}}\times {{x}^{c-a}}=P$.
$\Rightarrow {{x}^{a-c+c-a}}=P$.
$\Rightarrow {{x}^{0}}=P$ ---(1).
From the law of exponents, we know that ${{a}^{0}}=1$. We use this in equation (1) to get the value of P.
So, we get $P=1$ ---(2).
From equations (1) and (2), we get the value of P as ${{x}^{0}}$ or 1.
∴ The value of P is ${{x}^{0}}$ or 1.

So, the correct answer is “Option b and c”.

Note: We should not stop solving the problem after finding the answer ${{x}^{0}}=P$ as the problem can have multiple correct answers. We should check every answer present in the options while solving this problem. We should solve this problem step – by – step carefully without making any mistakes. Whenever we get problems involving the exponents of any power, we should solve it by using the laws of exponents as it makes the calculation easy.