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If $x, y, z$ are positive real numbers and $a,b,c$ are rational numbers, then the value of $\dfrac{1}{{1 + {x^{b - a}} + {x^{c - a}}}} + \dfrac{1}{{1 + {x^{a - b}} + {x^{c - b}}}} + \dfrac{1}{{1 + {x^{b - c}} + {x^{a - c}}}}$ is
A. $ - 1$
B. $0$
C. $1$
D. None of these

Answer Verified Verified
Hint: In this problem, we will find the required value by using the law of exponents. We can write ${x^{m - n}}$ as ${x^{m - n}} = \dfrac{{{x^m}}}{{{x^n}}}$. This is called the law of exponents. After using the law of exponents, we will simplify the expression to get the required value.

Complete step by step solution: In this problem, it is given that $x,y,z$ are positive real numbers and $a,b,c$ are rational numbers. We need to find the value of $\dfrac{1}{{1 + {x^{b - a}} + {x^{c - a}}}} + \dfrac{1}{{1 + {x^{a - b}} + {x^{c - b}}}} + \dfrac{1}{{1 + {x^{b - c}} + {x^{a - c}}}}\; \cdots \cdots \left( 1 \right)$.
Now we are going to use the law of exponents in expression $\left( 1 \right)$. that is, we are going to use the law ${x^{m - n}} = \dfrac{{{x^m}}}{{{x^n}}}$ in that expression. Therefore, we get
$\dfrac{1}{{1 + {x^{b - a}} + {x^{c - a}}}} + \dfrac{1}{{1 + {x^{a - b}} + {x^{c - b}}}} + \dfrac{1}{{1 + {x^{b - c}} + {x^{a - c}}}}$
$ = \dfrac{1}{{1 + \dfrac{{{x^b}}}{{{x^a}}} + \dfrac{{{x^c}}}{{{x^a}}}}} + \dfrac{1}{{1 + \dfrac{{{x^a}}}{{{x^b}}} + \dfrac{{{x^c}}}{{{x^b}}}}} + \dfrac{1}{{1 + \dfrac{{{x^b}}}{{{x^c}}} + \dfrac{{{x^a}}}{{{x^c}}}}}$
Let us simplify the above expression by taking LCM (least common multiple). Therefore, we get $\dfrac{1}{{\dfrac{{{x^a} + {x^b} + {x^c}}}{{{x^a}}}}} + \dfrac{1}{{\dfrac{{{x^b} + {x^a} + {x^c}}}{{{x^b}}}}} + \dfrac{1}{{\dfrac{{{x^c} + {x^b} + {x^a}}}{{{x^c}}}}}$
$ = \dfrac{{{x^a}}}{{{x^a} + {x^b} + {x^c}}} + \dfrac{{{x^b}}}{{{x^a} + {x^b} + {x^c}}} + \dfrac{{{x^c}}}{{{x^a} + {x^b} + {x^c}}}$
Let us rewrite the above expression by taking LCM. Therefore, we get
$\dfrac{{{x^a} + {x^b} + {x^c}}}{{{x^a} + {x^b} + {x^c}}}$
On cancellation of the term ${x^a} + {x^b} + {x^c}$, we get the required value and it is $1$.
Therefore, if $x,y,z$ are positive real numbers and $a,b,c$ are rational numbers, then the value of $\dfrac{1}{{1 + {x^{b - a}} + {x^{c - a}}}} + \dfrac{1}{{1 + {x^{a - b}} + {x^{c - b}}}} + \dfrac{1}{{1 + {x^{b - c}} + {x^{a - c}}}}$ is equal to $1$.
Therefore, option C is correct.

Note: A rational number is of the form $\dfrac{p}{q}$ where $p$ and $q$ are integers. Note that here $q$ is non-zero. In this type of problem, laws of exponents are very useful to find the required value. In some problems, we can use the law ${x^{m + n}} = {x^m} \times {x^n}$. In some problems, we can use the law ${\left( {{x^m}} \right)^n} = {x^{m\; \times \;n}}$. These are called laws of exponents. In the term ${x^m}$, we can say that $x$ is the base and $m$ is the exponent. Exponent is also known as power or index. Exponent of any number says how many times we need to multiply that number. For example, in the term ${2^8}$, exponent is $8$. So, we can say that we need to multiply the number $2$ itself $8$ times to find the value of ${2^8}$.