Question

What is the cube root of \[{{x}^{8}}\]?

Answer 1

Hint: From the given question we have been asked to find the cube root of \[{{x}^{8}}\]. To solve this question we will take the cube root of a number definition and then we apply the cube root form to the given number. Later we will simplify the expression using the basic mathematical operations like addition, subtractions and multiplications and then we will write the answer in reduced form. So, the solution will be as follows.

Complete step-by-step answer:
Generally, in mathematics the term cube root of a number is a value which when multiplied by it thrice or simply which when multiplied by itself three times produces the original value.
In mathematical notation let there be a number \[a\] then the cube root of \[a\] will be \[\sqrt[3]{a}\].
For, the given question that is the cube root of \[{{x}^{8}}\] will be as follows.
 \[= \sqrt[3]{{{x}^{8}}}\]
Now we will expand the term inside the cube root as follows.
\[= \sqrt[3]{{{x}^{2}}\times {{x}^{6}}}\]
Now, again ${{x}^{6}}$can be written as ${{\left( {{x}^{2}} \right)}^{3}}$ by this we will get,
$= \sqrt[3]{{{x}^{2}}\times {{\left( {{x}^{2}} \right)}^{3}}}$
So, the term inside the cube root ${{\left( {{x}^{2}} \right)}^{3}}$ will come out of the cube root as follows.
\[= {{x}^{2}}\sqrt[3]{{{x}^{2}}}\]
This expression can’t be simplified.
Therefore, the solution for the given question will be \[= {{x}^{2}}\sqrt[3]{{{x}^{2}}}\].

Note: Students must be very careful in doing the calculations. Students should have good knowledge in the concept of cube root of a number or a variable. We can also solve this question in the form of fractional powers as follows.
\[= \sqrt[3]{{{x}^{8}}}\]
\[= \sqrt[3]{{{x}^{2}}\times {{x}^{6}}}\]
this can be written as,
$= {{x}^{\dfrac{2}{3}}}\times {{x}^{\dfrac{6}{3}}}$
\[= {{x}^{2}}\sqrt[3]{{{x}^{2}}}\]