The problem asks to simplify the given expression: $(\frac{2r^{-3}t^6}{5u^9})^4$

1. Problem Description

The problem asks to simplify the given expression:

$(\frac{2r^{-3}t^6}{5u^9})^4$

2. Solution Steps

We will use the power of a quotient rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$

$(\frac{2r^{-3}t^6}{5u^9})^4 = \frac{(2r^{-3}t^6)^4}{(5u^9)^4}$

Now, use the power of a product rule: $(ab)^n = a^n b^n$

$\frac{(2r^{-3}t^6)^4}{(5u^9)^4} = \frac{2^4 (r^{-3})^4 (t^6)^4}{5^4 (u^9)^4}$

We have $2^4 = 16$ and $5^4 = 625$. Use the power of a power rule: $(a^m)^n = a^{mn}$.

Thus we have:

$\frac{16 r^{-3*4} t^{6*4}}{625 u^{9*4}} = \frac{16 r^{-12} t^{24}}{625 u^{36}}$

We want to express the answer with positive exponents. Use the rule $a^{-n} = \frac{1}{a^n}$.

So, $r^{-12} = \frac{1}{r^{12}}$.

$\frac{16 r^{-12} t^{24}}{625 u^{36}} = \frac{16 t^{24}}{625 r^{12} u^{36}}$

3. Final Answer

$\frac{16t^{24}}{625r^{12}u^{36}}$