Question

How do you write the given term ${{5}^{\dfrac{2}{3}}}$ in radical form?

Answer 1

Hint: We start solving the problem by recalling the definition of simplest radical form as expressing the given radical to a form in which there is no further simplification of roots can be performed. We then make use of the law of radicals that ${{a}^{\dfrac{m}{n}}}={{\left( {{a}^{m}} \right)}^{\dfrac{1}{n}}}$ to proceed through the problem. We then make use of the law of radicals that ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$ to get the required answer for the given problem.

Complete step-by-step answer:
According to the problem, we are asked to write the given term ${{5}^{\dfrac{2}{3}}}$ in radical form.
Let us assume $r={{5}^{\dfrac{2}{3}}}$ ---(1).
Let us recall the definition of simplest radical form.
We know that the simplest radical form is expressing the given radical to a form in which there is no further simplification of roots can be performed.
From the laws of radicals, we know that ${{a}^{\dfrac{m}{n}}}={{\left( {{a}^{m}} \right)}^{\dfrac{1}{n}}}$. Let us use this result in equation (1).
$\Rightarrow r={{\left( {{5}^{2}} \right)}^{\dfrac{1}{3}}}$.
$\Rightarrow r={{25}^{\dfrac{1}{3}}}$ ---(2).
From the laws of radicals, we know that ${{a}^{\dfrac{1}{n}}}=\sqrt[n]{a}$. Let us use this result in equation (2).
$\Rightarrow r=\sqrt[3]{25}$.
So, we have found the simplified radical form of ${{5}^{\dfrac{2}{3}}}$ as $\sqrt[3]{25}$.
$\therefore $ The simplified radical form of ${{5}^{\dfrac{2}{3}}}$ is $\sqrt[3]{25}$.

Note: Whenever we get this type of problem, we first try to recall the required definition to get the required answer. We should keep in mind that the terms present inside the root cannot be written as $\sqrt[n]{a}=\sqrt[n]{{{b}^{n}}\times c}$ while solving this type of problems. We should confuse ${{a}^{\dfrac{1}{n}}}$ with ${{a}^{n}}$ instead of $\sqrt[n]{a}$, which is the common mistake done by students. Similarly, we can expect problems to write the following terms into simplest radical form: ${{64}^{\dfrac{1}{4}}}$, ${{343}^{\dfrac{1}{2}}}$, ${{243}^{\dfrac{1}{3}}}$.