Question

How do you simplify ${\left( {64} \right)^{\dfrac{1}{6}}}$?

Answer 1

Hint: Here we need to find the value of ${\left( {64} \right)^{\dfrac{1}{6}}}$ so this means that we need to actually find the number which when multiplied $6$ times by itself the result must be $64$ so we need to find that number.

Complete step by step solution:
Here we are given to find the value of ${\left( {64} \right)^{\dfrac{1}{6}}}$
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 ${\left( {64} \right)^{\dfrac{1}{6}}}$.
As we know that ${a^{\dfrac{m}{n}}} = {({a^{\dfrac{1}{n}}})^m}$
Hence we can also write ${\left( {64} \right)^{\dfrac{1}{6}}}$ as ${\left( {{{64}^{\dfrac{1}{2}}}} \right)^{\dfrac{1}{3}}}$.
Hence we can first find the value of ${\left( {64} \right)^{\dfrac{1}{2}}}$ and we now that here we need to find the number that is multiplied $2$ times and we get the value $64$
Hence we know that
$\left( 8 \right)\left( 8 \right) = 64{\text{ and }}\left( { - 8} \right)\left( { - 8} \right) = 64$
Hence we can say that ${\left( {64} \right)^{\dfrac{1}{2}}} = \pm 8$
Now we just need to put its value in ${\left( {{{64}^{\dfrac{1}{2}}}} \right)^{\dfrac{1}{3}}}$ and we will get ${\left( 8 \right)^{\dfrac{1}{3}}}$ and hence we need to find the number that is multiplied three times gives $8$ so we know that $\left( 2 \right)\left( 2 \right)\left( 2 \right) = 8$
So we come to know that ${\left( {64} \right)^{\dfrac{1}{6}}} = 2$

Also we have got that ${\left( {64} \right)^{\dfrac{1}{2}}} = - 8$ hence we can also find the value of ${\left( { - 8} \right)^{\dfrac{1}{3}}}$ and hence we need to find the number that is multiplied three times gives $8$ so we know that $\left( { - 2} \right)\left( { - 2} \right)\left( { - 2} \right) = - 8$.

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 ${a^{\dfrac{m}{n}}} = {({a^{\dfrac{1}{n}}})^m}$.