Question

How do you simplify ${\left( { - 243} \right)^{\dfrac{3}{5}}}$?

Answer 1

Hint: Here we can divide the question into two parts. First we can find the value of ${\left( { - 243} \right)^{\dfrac{1}{5}}}$ and this means a number which when multiplied $5$ times gives us the result $ - 243$.
Then the resultant number can be multiplied three times to find the cube of that number and we will get ${\left( { - 243} \right)^{\dfrac{3}{5}}}$.

Complete step by step solution:
Here we are given to find the value of ${\left( { - 243} \right)^{\dfrac{3}{5}}}$
So we know when we are given to find the value of ${\left( 4 \right)^{\dfrac{1}{2}}}$ this means that we need to find the number which when multiplied $2$ times gives the result as $4$ and we know that when $2{\text{ or }} - 2$ is multiplied two times the result is $4$.
Hence we can say that ${\left( 4 \right)^{\dfrac{1}{2}}} = \pm 2$
Here also we are given such a type of problem where we need to find the value of ${\left( { - 243} \right)^{\dfrac{3}{5}}}$.
As we know that ${a^{\dfrac{m}{n}}} = {({a^{\dfrac{1}{n}}})^m}$
Hence we can also write ${\left( { - 243} \right)^{\dfrac{3}{5}}}$ as ${\left( { - {{243}^{\dfrac{1}{5}}}} \right)^3}$.
Hence we can first find the value of ${\left( { - 243} \right)^{\dfrac{1}{5}}}$ and we now that here we need to find the number that is multiplied $5$ times and we get the value $ - 243$.
We know that this will be a negative number as a positive number when multiplied any number of times cannot give us the results as negative.
Hence we know that
$
  ( - 2)( - 2)( - 2)( - 2)( - 2) = - 32 \\
  \left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right) = - 243 \\
 $
Hence we can say that ${\left( { - 243} \right)^{\dfrac{1}{5}}} = - 3$

Now we just need to put its value in ${\left( { - {{243}^{\dfrac{1}{5}}}} \right)^3}$ and we will get $ - {3^3} = \left( { - 3} \right)\left( { - 3} \right)\left( { - 3} \right) = - 27$

Note:
Here whenever we are given such a problem we must keep in mind that whenever we are given the power in fraction we can split that because of the property which states that ${a^{\dfrac{m}{n}}} = {({a^{\dfrac{1}{n}}})^m}$.