Question

What is the radical form for ${{16}^{\dfrac{5}{4}}}$ ?

Answer 1

Hint: We are asked to write the radical form of the given number in the question. So, for that we need to be aware of power and exponents and also with the fractional power of numbers and then writing it as the root.

Complete step-by-step answer:
In the given question, simply we are asked to write the radical form of the given number which is raised to fractional power $\dfrac{5}{4}$ .
Now, in order to get the radical form of this we need to use the root property and power distribution or we can say arrangement like we can write this as ${{\left( {{16}^{5}} \right)}^{\dfrac{1}{4}}}$ . So, we can write this term as one fourth root of the number ${{16}^{5}}$ .
It will be written as $^{4}\sqrt{{{16}^{5}}}$ .
Now, as it can be solved by making factors inside the root and can make the fair of numbers. Now, as soon as we get the pair of four terms of the same number then that number will come out of the root and the remaining number can be simplified further by their prime factors in order to get simplified if the number left inside the root is larger.
So, for the given question we have $^{4}\sqrt{{{16}^{5}}}$, so for finding the actual value of this term we will get five 16 inside the root and making a pair of 4, 16’s we can bring 16 outside the root and we will be left with one 16 inside the root and that be written as $2\times 2\times 2\times 2$ . So now we can bring 2 outside the root and hence the actual value we get is 32.
Like mathematically we can see
$\begin{align}
  & {{\left( 16\times 16\times 16\times 16\times 16 \right)}^{\dfrac{1}{4}}} \\
 & \Rightarrow 16{{\left( 16 \right)}^{\dfrac{1}{4}}} \\
 & \Rightarrow 16\times 2 \\
 & \Rightarrow 32 \\
\end{align}$
Since,
$\begin{align}
  & {{2}^{4}}=16 \\
 & \Rightarrow {{({{2}^{4}})}^{\dfrac{1}{4}}} \\
 & \Rightarrow 2 \\
\end{align}$
Therefore, the radical form of the number given in the question is $^{4}\sqrt{{{16}^{5}}}$ or we can say the radical form of given number is 2.

Note: In this question is not very tough what we need to do is take care while bringing the numbers out of the root and then simplifying. Also, we must know how we need to write fractional power as the root of that denominator term.