We need to simplify the expression $(\frac{2r^3t^6}{5u^9})^4$.

1. Problem Description

We need to simplify the expression $(\frac{2r^3t^6}{5u^9})^4$.

2. Solution Steps

First, we apply the power of a quotient rule, which states that $(\frac{a}{b})^n = \frac{a^n}{b^n}$.

So, we have:

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

Next, we apply the power of a product rule, which states that $(ab)^n = a^n b^n$.

So, we have:

$(2r^3t^6)^4 = 2^4 (r^3)^4 (t^6)^4$

and

$(5u^9)^4 = 5^4 (u^9)^4$

Now, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$.

So, we have:

$(r^3)^4 = r^{3*4} = r^{12}$

$(t^6)^4 = t^{6*4} = t^{24}$

$(u^9)^4 = u^{9*4} = u^{36}$

Substituting these back into the expression, we get:

$\frac{(2r^3t^6)^4}{(5u^9)^4} = \frac{2^4 r^{12} t^{24}}{5^4 u^{36}}$

We know that $2^4 = 16$ and $5^4 = 625$.

So, we have:

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

3. Final Answer

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