Fractional Exponents powers

Question

I am having problems understanding how to answer questions containing fractional exponents to a given power ie $(2x^{1/2})^6$, i do not understand how to go about answering the question. I know this is an easy topic, but i would really appreciate the help

Answer 1

The general rule for exponents is $x^{a/b} = \sqrt[b]{x^a} \forall x \in \mathbb{N}$. For example, $4^{2/3} = \sqrt[3]{4^2} = \sqrt[3]{8}$. You can apply this to your problem to see that $(2x^{1/2})^6 = 64x^3.$

Answer 2

how to answer questions containing fractional exponents

In the exact same manner in which you answer questions containing non-fractional exponents, since the base is obviously a positive number, given the fact that both $2$ and $\sqrt x$ are always $\ge0$. $(ab)^c=a^cb^c$ for all exponents, fractional or not, it doesn't matter. Likewise, $(a^b)^c=a^{bc}$, again, for all exponents. By combining the two, we have $(ab^c)^d=a^db^{cd}$. In this case, $a=2,b=x,c=$ $=\dfrac12$ and $d=6$. And $\dfrac12\cdot6=3$.