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: We need to find the value of $x$ in the given equation. We will be solving the given problem using the product of powers rule in exponents. According to the rule, we add the exponents of the terms with the same bases. Then, we simplify the expression to get the desired result.

Complete step by step answer:
We are given an equation and need to find the value of $x$ in it. We will be solving the given question using the rules of exponents.
Exponents, in mathematics, refer to the number of times a number is multiplied by itself. It is written above the base number to the right.
The general representation of the exponents is given by
$\Rightarrow {{a}^{x}}=a\times a\times a.....\times a$
Here,
$a=$ base
$x=$ exponent
The product of powers rule in exponents is used when we are multiplying two terms with the same base.
It states that when we are multiplying two bases of the same value then we need to keep the bases the same and just add the powers to get the result. It is given as follows,
$\Rightarrow {{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}$
Let us understand the working of the product of power rule with an example,
Example:
Find the value of ${{2}^{2}}\times {{2}^{3}}?$
In the given question, the bases of both terms are the same.
The result is obtained just by adding the corresponding exponents of the terms.
Applying the same, we get,
$\Rightarrow {{2}^{\left( 2+3 \right)}}$
$\Rightarrow {{2}^{5}}$
$\therefore {{2}^{2}}\times {{2}^{3}}={{2}^{5}}$
According to our question,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-15}}\times{{\left( \dfrac{5}{3} \right)}^{11}}={{\left( \dfrac{5}{3} \right)}^{8x}}$
On the left-hand side of the equation, the bases of the terms are the same.
Applying the product of powers rule on the left side of the equation, we get,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-15+11}}={{\left( \dfrac{5}{3} \right)}^{8x}}$
Simplifying the above equation, we get,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-4}}={{\left( \dfrac{5}{3} \right)}^{8x}}$
From the rules of exponents, when bases are the same, powers can be equated.
Following the same, we get,
$\Rightarrow -4=8x$
$\Rightarrow x=\dfrac{-4}{8}$
$\therefore x=\dfrac{-1}{2}$

Note: The result of the given question can be cross-checked using the equation ${{\left( \dfrac{5}{3} \right)}^{-15}}\times{{\left( \dfrac{5}{3} \right)}^{11}}={{\left( \dfrac{5}{3} \right)}^{8x}}$ .
LHS:
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-15}}\times{{\left( \dfrac{5}{3} \right)}^{11}}$
Applying the product of power rule,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-4}}$
RHS:
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{8x}}$
Substituting the value of $x$ , we get,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{8\left( \dfrac{-1}{2} \right)}}$
Simplifying the above expression, we get,
$\Rightarrow {{\left( \dfrac{5}{3} \right)}^{-4}}$
LHS $=\;$ RHS
The result attained is correct.