I am having a small problem recalling how to factor with exponents and roots.
For example, I understand $\sqrt{16t^2+4t^4}$=$2t\sqrt{4+t^2}$
But I have issues when it is factoring not with a square root, but say a (3/2) for example.
For instance , in my book it writes $$(4e^{2t}+4e^{4t})^{3/2}=8e^{3t}(1+e^{2t})^{3/2}$$
And I don't see exactly how it is done. So I am looking for tips in general on how to understand this, and for factoring out with roots in general. Thanks
I think the thing to notice, when looking at $(4e^{2t} + 4e^{4t})^{3/2}$, is that the two terms $4e^{2t}$ and $4e^{4t}$ have a common factor: each is a multiple of $4e^{2t}$. So one can extract that common factor as follows: $$\begin{align} (4e^{2t} + 4e^{4t})^{3/2}& = (4e^{2t}(1+e^{2t}))^{3/2} \\ & = (4e^{2t})^{3/2}(1+e^{2t})^{3/2} \\ & = 4^{3/2}(e^{2t})^{3/2}(1+e^{2t})^{3/2} \\ & = 8\cdot e^{3t}\cdot (1+e^{2t})^{3/2} \end{align}$$
(If any part of this wasn't clear, please leave a comment, and I will explain.)
Trying a similar thing with $\sqrt{16t^2 + 4t^4}$ also works: the two terms $16t^2$ and $4t^4$ have a common factor of $4t^2$, so let's extract that:
$$\begin{align} \sqrt{16t^2 + 4t^4}& = \sqrt{(4t^2)(4+t^2)}\\ &= \sqrt{4t^2}\sqrt{4+t^2} \\ &= 2t\sqrt{4+t^2} \end{align}$$