How do you get from this to this formula?

Question

I have the formula : $$3×4^{n-1}×1×\left({1\over 3}\right)^{n-1}$$ And I would like to know how to get to this one (which is equal) : $$3× \left({4\over 3}\right)^{n-1}$$ How can I do that ?

Answer 1

Well first you can rewrite

$(3)$ x $(4^{n-1})$ x $1$ x $({{1}\over {3}})^{n-1}$ as

$(3)$ x $(4^{n-1})$ x $({{1}\over {3}})^{n-1}$.

Then using the identity that $a^nb^n = (ab)^{n}$ we get

$(3)$ x $({{4}\over {3}})^{n-1}$

which is the answer you wanted.

Answer 2

Hint: Use the fact that $a^nb^n = (ab)^n$. Put $a = 4,\;b = \frac 13$.