Question

If ${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$, then $x = ?$

Answer 1

Hint:
In this question, we have to use the law of exponents which states that when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. For any real numbers $a$, $x$ and $y$, where \[a > 0\], ${a^x} = {a^y}$ if and only if $x = y$.

Complete step by step solution:
Given, ${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$ …..(1)
The expression on LHS i.e., ${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}}$ can be solved by using product rule of exponents, which states that when multiplying two powers that have the same base, we can add the exponents.
$\therefore $${\left( {\dfrac{5}{3}} \right)^{ - 15 + 11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$ $\left( {\because {a^m} \cdot {a^n} = {a^{m + n}}} \right)$
${\left( {\dfrac{5}{3}} \right)^{ - 4}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$ ….. (2)
As we know that when an exponential equation has the same base on each side, the exponents must be equal.
Since in equation (2), the base on both sides is the same i.e., $\left( {\dfrac{5}{3}} \right)$. Therefore, the exponents also are equal.
$ \Rightarrow 8x = - 4$
$ \Rightarrow x = \dfrac{{ - 4}}{8}$
$ \Rightarrow x = \dfrac{{ - 1}}{2}$

Therefore, the value of $x$ is $\dfrac{{ - 1}}{2}$.

Note:
While solving this type of questions, it may be noted that when multiplying two powers that have the same base, we can add the exponents. It is called the product rule of exponents. For any real numbers $a$, $x$ and $y$, where \[a > 0\];
${a^x} \cdot {a^y} = {a^{x + y}}$