Question

What is the simplified radical form of ${{\left( 5 \right)}^{\dfrac{5}{2}}}$ ?

Answer 1

Hint: In this question we have to find the radical form of the given expression. Thus we will use the basic mathematical rule and the square root rule to get the solution. As we know that radical means to change a number in square root form, that is $\sqrt{{}}$. So, in this problem we will first separate $\dfrac{5}{2}$ as $\dfrac{1}{2}$ and 5 and then we will put square root at place of $\dfrac{1}{2}$ and now the power of 5 will be 5. Then we can take ${{5}^{5}}$ as ${{5}^{4}}.5$ and ${{5}^{4}}$ will go outside from the under root.

Complete step by step solution:

According to the problem, we have to get the radical form from the given expression. Thus we will use the square root rule and the basic mathematical rule to get the solution. The expression given to us is ${{\left( 5 \right)}^{\dfrac{5}{2}}}$. Now if we write $\dfrac{5}{2}$ as multiplication of $\dfrac{1}{2}$ and 5, then we get,

${{5}^{5\times \dfrac{1}{2}}}$

Now we will remove $\dfrac{1}{2}$ in the power, by putting square root symbol at that place, that is $\sqrt{{}}$ outside ${{5}^{5}}$, thus we get,

$\sqrt{{{5}^{5}}}$

Now again we will simplify the square root by taking ${{5}^{4}}$ out from the square root because ${{5}^{4}}$ can be written as $5\times 5\times 5\times 5$. Therefore, we will make the group of two multiplications of 5s. So, we get,

 $5\times 5\text{ }\times \text{ }5\times 5$

Thus, when we take ${{5}^{4}}$ from the square root, we get two 5 from the same, therefore, we get,

${{5}^{2}}\sqrt{5}$ , which is the required solution $25\sqrt{5}$.

Therefore, for the expression ${{\left( 5 \right)}^{\dfrac{5}{2}}}$ , the radical form is equal to $25\sqrt{5}$.

Note: To avoid confusion, make a thorough note of all the procedures when answering this question. You can also solve your solution after the $25\sqrt5$ step. Remember to include the definition of radial form because the solution must include a square root symbol.