Question

How do you write ${x^{\dfrac{1}{2}}}$ in radical form?

Answer 1

Hint: The reciprocal of the number associated with the radical is the power needed. For example, $\sqrt[n]{x}$can be written in a similar way as, ${x^{\dfrac{1}{n}}}$.

Complete step by step solution:
Firstly we look at an expression of the form: ${x^{\dfrac{1}{n}}}$.
To investigate what this means, we need to go from $x \to {x^{\dfrac{1}{n}}}$ and then try to deduce something from it.
We get that, ${x^1}$$ = {x^{\dfrac{n}{n}}}$$ = {x^{n.\dfrac{1}{n}}}$
Now, we still know that this number is equal to $x$. So now we need to think carefully a bit about what number, multiplied by itself n times, gives you x.

We can deduce that, it's the nth-root of $x \Rightarrow $ ${x^{\dfrac{1}{n}}}$$ = \sqrt[n]{x}$
According to the given information we need to write ${x^{\dfrac{1}{2}}}$in radical form.
So, we derived that, ${x^{\dfrac{1}{2}}}$ is defined as the ${2^{nd}}$root of $x$.

Therefore, ${x^{\dfrac{1}{2}}}$is equivalent to $\sqrt[2]{x}$.

Note:
You can just remember this rule which depicts,
${x^{\dfrac{a}{b}}} = \sqrt[b]{{{x^a}}} = {(\sqrt[b]{x})^a}$.