Question

The value of ${{\left( 243 \right)}^{\dfrac{-2}{5}}}$ is
A. $\dfrac{1}{9}$
B. $\dfrac{2}{9}$
C. $9$
D. None of these

Answer 1

Hint: We are asked to find the value of the expression ${{\left( 243 \right)}^{\dfrac{-2}{5}}}$ . We start to solve the given question by expressing the number 243 to the power of 3. Then, we use the rules of exponents to get the desired result.

Complete step by step solution:
We are given an expression and are asked to find the value of it. We will be solving the given question using the rules of exponents.
An exponent of a given number refers to the number of times a number is multiplied by itself. It is given as follows,
$\Rightarrow {{a}^{x}}=a\times .........\times a$
The number a is multiplied x times in the above equation.
In the above equation,
x is the exponent
a is the base
According to our question, we need to find the value of the expression ${{\left( 243 \right)}^{\dfrac{-2}{5}}}$ .
We need to express the number 243 to the power of the number 3.
Following the same, we get,
$\Rightarrow 243=3\times 3\times 3\times 3\times 3$
In the above equation, 5 times the number 3 is multiplied by itself
Expressing the above equation in the form of exponent, we get,
$\therefore 243={{3}^{5}}$
Substituting the value of the number 243 in the given expression, we get,
$\Rightarrow {{\left( 243 \right)}^{\dfrac{-2}{5}}}={{\left( {{3}^{5}} \right)}^{\dfrac{-2}{5}}}$
From the rules of exponents, we know that
$\Rightarrow {{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
In our case,
m = 5
$n=\dfrac{-2}{5}$
Applying the rule of exponents to the above equation, we get,
$\Rightarrow {{\left( 243 \right)}^{\dfrac{-2}{5}}}={{3}^{\dfrac{-2}{5}\times 5}}$
Simplifying the above equation, we get,
$\Rightarrow {{\left( 243 \right)}^{\dfrac{-2}{5}}}={{3}^{-2}}$
The value of ${{3}^{-2}}$ is given as follows,
$\Rightarrow {{3}^{-2}}=\dfrac{1}{{{3}^{2}}}=\dfrac{1}{9}$
Substituting the same, we get,
$\therefore {{\left( 243 \right)}^{\dfrac{-2}{5}}}=\dfrac{1}{9}$

So, the correct answer is “Option A”.

Note: We must know the basic rules of the powers and the exponents to solve the given question in less time. We must remember that the value of the expression ${{a}^{-n}}$ is exactly the same as the value of $\dfrac{1}{{{a}^{n}}}$ .