Question

How do you write ${{4}^{\dfrac{3}{2}}}$ in radical form?

Answer 1

Hint: The radical form is the representation of a number in which there is at least one radical sign. The radical sign is either the square root or any higher order root. We will use the identity ${{x}^{ab}}={{\left( {{x}^{a}} \right)}^{b}}.$ We will also use another identity $\sqrt{ab}=\sqrt{a}\sqrt{b}.$

Complete step by step answer:
Let us consider the number ${{4}^{\dfrac{3}{2}}}.$
We need to write this number in the radical form.
The radical form of a number is a representation of a number that includes the radical sign.
Let us first consider the exponent of the given number. It is $\dfrac{3}{2}.$
We can write this fraction as a product of two numbers. And this will be done as $\dfrac{3}{2}=3\times \dfrac{1}{2}.$
Let us substitute for the exponent in the given number.
Then, the given number becomes ${{4}^{\dfrac{3}{2}}}={{4}^{3\times \dfrac{1}{2}}}.$
This is in the form ${{x}^{ab}}.$
So, we can use a familiar identity to reach the next stage of the conversion.
We will use the identity ${{x}^{ab}}={{\left( {{x}^{a}} \right)}^{b}}.$
Here, we will take $a=3$ and $b=\dfrac{1}{2}.$
So, we will get ${{4}^{\dfrac{3}{2}}}={{\left( {{4}^{3}} \right)}^{\dfrac{1}{2}}}.$
From this, we will get the next step in which we expand the terms inside the brackets. Thus, we will get ${{4}^{\dfrac{3}{2}}}={{\left( 4\times 4\times 4 \right)}^{\dfrac{1}{2}}}.$
We know that $4\times 4\times 4$ is equal to $64.$ That is, the cube of $4$ is $64.$
Therefore, we will get ${{4}^{\dfrac{3}{2}}}={{64}^{\dfrac{1}{2}}}.$
If the exponent of a number is $\dfrac{1}{2},$ then that means the square root of the number.
We know that the square root of $64$ is $8.$ We can confirm this by ${{4}^{\dfrac{3}{2}}}=\sqrt{4\times 4\times 4}=\sqrt{16\times 4}=\sqrt{16}\sqrt{4}=4\times 2=8.$ We have used the identity $\sqrt{ab}=\sqrt{a}\sqrt{b}.$
Thus, we will get ${{4}^{\dfrac{3}{2}}}=8.$
So, this number does not contain the radical sign.
Hence the given number ${{4}^{\dfrac{3}{2}}}$ does not have radical form.

Note:
We can understand that we can write a number in the radical form if the term under the square root is not a perfect square. If the radical sign is the cube root, then the term under it should not be a perfect cube and so on.