How do you write ${{\sqrt[3]{10}}^{7}}$ in rational exponential notation?

Hint: Here in this question we have been asked to write the given mathematical expression ${{\sqrt[3]{10}}^{7}}$ in rational exponential notation. From the basic concepts of algebra we know that we can write the mathematical expression $\sqrt[n]{a}$ as ${{a}^{\dfrac{1}{n}}}$ in rational exponential notation.

Complete step by step answer:
Now considering from the question we have been asked to write the given mathematical expression ${{\sqrt[3]{10}}^{7}}$ in rational exponential notation.
From the basic concepts of algebra we know that we can write the mathematical expression $\sqrt[n]{a}$ as ${{a}^{\dfrac{1}{n}}}$ in rational exponential notation. Now we will use this in the process of our solution.
By using this concept we will have ${{\sqrt[3]{10}}^{7}}$ as $\Rightarrow {{\left( {{10}^{\dfrac{1}{3}}} \right)}^{7}}$ .
This can be further simplified and written as $\Rightarrow {{10}^{\dfrac{7}{3}}}$ .
Therefore we can conclude that the given mathematical expression ${{\sqrt[3]{10}}^{7}}$ can be written as ${{10}^{\dfrac{7}{3}}}$ in rational exponential form.

Note: During the process of answering questions of this type we should be sure with the concepts that we are going to apply. This is a very simple and easy question and can be answered accurately in a short span of time. Very few mistakes are possible in questions of this type. If someone had confused with the notation and written it as $\sqrt[n]{a}={{a}^{n}}$ then we will have ${{\sqrt[3]{10}}^{7}}={{\left( {{10}^{3}} \right)}^{7}}\Rightarrow {{10}^{21}}$ which is a wrong answer clearly. So we should be careful with the concepts that we are going to apply in questions of this type.