Question

How do you take the cube root of an exponent?

Answer 1

Hint: First take an exponential number and apply cube root to it and then use the law of indices for fractional powers to express the cube root with the exponent of the considered exponential number.
Law of indices for fractional power for “a” raise to the power “b divided by c” is given as
${a^{\dfrac{b}{c}}} = \sqrt[c]{{{a^b}}}$
Use this formula to express the cube root in the exponent.

Complete step by step solution:
Let us take a number (say $x$) which is raised to the power of some other number (say $a$). Therefore the exponential number will look like the following:
$ = {x^a}$
Now the cube root of this exponential number will be given as
$ = \sqrt[3]{{{x^a}}}$
From the law of indices for fractional powers we know that
$\sqrt[c]{{{a^b}}} = {a^{\dfrac{b}{c}}}$
Using this to express cube root with the exponent of the considered exponential number
$ \Rightarrow \sqrt[3]{{{x^a}}} = {x^{\dfrac{a}{3}}}$
Therefore cube root of an exponent can be taken as the power equals to the division of the given exponent with $3$.

Additional Information:
Physical significance of cube root could be understood by the length of a side of the cube whose volume is equal to the cube of the length of the sides of the cube.

Note: Cube root of a number gives the number which when multiplied by itself two times gives the number whose cube root is taken. Say if the cube root of the number $k$ equals the number $j$ then we can write $j \times j \times j = k$ . Cube and cube root are inverse operations to each other.