Simplify an expression involving indices

Question

One of my friends asked me this question: Simplify $$\frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}$$ I've been thinking about the question for more than a day. I've looked through my teacher's notes but none of the examples my teacher gave is similar to this question. It's from Form 4 syllabus, Chapter 5: Logarithms and Indices. Any help would be much appreciated. Thanks :)

Answer 1

Hint: Two approaches: 1)If you write $250^{3x+1}=250^2\cdot 250^{3x-1}$ you can cancel some things. 2)Note that you only have factors of $2$ and $5$. Break up $50,10,250$ into $2$'s and $5$'s and find the exponent for each. I think the second is what is intended, but the first is what hit me first and it may help in other situations.

Answer 2

From the second hint of the first answer; \begin{equation*} \begin{split} \frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}& = \frac{5^{3x}\times2^1}{5^{3(3x+1)}\times2^{3x+1}}\\ & = 5^{-6x-3}\times 2^{-3x}\\ & =(5^{2})^{-3x}\times 2^{-3x}\times 5^{-3}\\ & =25^{-3x}\times 2^{-3x}\times 125^{-1}\\ & =\frac{50^{-3x}}{125}\\ \end{split} \end{equation*}