Question

How do you write \[{21^{\dfrac{9}{4}}}\] in radical form?

Answer 1

Hint: Here radical form is nothing but root form. Like square root , cube root. And then we will use the laws of indices and powers. As we know that any number of the form \[{a^{\dfrac{x}{n}}}\] can be written in the radical form as \[\sqrt[n]{{{a^x}}}\]. So this is our hint to lead in this problem.

Complete step-by-step answer:
We are given that \[{21^{\dfrac{9}{4}}}\] is to be written in radical form.
Now we know that \[{a^{\dfrac{1}{n}}} = \sqrt[n]{a}\]
But here in the case above we have a rational exponent or we can say an exponent in fraction form.
So we will use the formula mentioned above.
\[{a^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}}\]
Here x is 9 and n is 4.
So we can write
\[{21^{\dfrac{9}{4}}} = \sqrt[4]{{{{21}^9}}}\]
This is the answer.
So, the correct answer is “$ \sqrt[4]{{{{21}^9}}}$”.

Note: Note that when we solve the problems related to exponents and powers the laws are used according to the conditions given. Here we are given a number with base and its exponent is in the form of a rational number. In this case a specified formula is used. This type of solution is generally used in simplifying the roots or writing the roots in its simplified form.