Question

How do you simplify ${{64}^{\dfrac{1}{4}}}$?

Answer 1

Hint: For solving this question, we have to learn about the power of the coefficient. Like ${{2}^{2}}=4$, ${{2}^{3}}=8$ and so on. Then, after simplifying the given problem statement you will obtain the result. So, let’s have a look at how we can approach the given question.

Complete Step by Step Solution:
The given problem to simplify is ${{64}^{\dfrac{1}{4}}}$.
Now, we will convert the given number 64 to the power for that it will be 2 to the power 6 which when multiplied together will give us the required number, that means, we get,
$\Rightarrow 64={{2}^{6}}$
Now put the value in the above question, that means, we get,
$={{({{2}^{6}})}^{\dfrac{1}{4}}}$
Now, by the law of exponent that means here we will use the power of a power rule which states that we need to multiply the exponents, by doing the process, we get,
$={{2}^{6\times \dfrac{1}{4}}}$
Now when we simplify it, we get,
$={{2}^{\dfrac{3}{2}}}$
Now, we will break ${{2}^{\dfrac{3}{2}}}$ down into its component parts$\dfrac{3}{2}$ is the same as $3\times \dfrac{1}{2}$ which is the same as $\dfrac{1}{2}\times 3$, that means, we get,
$={{2}^{3\times \dfrac{1}{2}}}$
And we know ${{2}^{3\times \dfrac{1}{2}}}$is $2\sqrt{2}$, that means, we get,

So, when we simplify ${{64}^{\dfrac{1}{4}}}$ we get $2\sqrt{2}$.

Note:
There is an alternative method for the problem ${{64}^{\dfrac{1}{4}}}$.
Let x = ${{64}^{\dfrac{1}{4}}}$
Now simply we will change $\dfrac{1}{4}$ in the form of 4th root, that means, we get,
$\Rightarrow x=\sqrt[4]{64}$
When we simplify it, we get,
$\Rightarrow x=\sqrt{8}$
As 8 is not a perfect square we will get the result as,
$\Rightarrow x=2\sqrt{2}$
So, when we simplify ${{64}^{\dfrac{1}{4}}}$ we get $2\sqrt{2}$.