To simplify the expression in the image, we follow these steps:

Given Expression:

$\frac{\frac{10t^3u}{3rs^3}}{\frac{5t^4u^4}{9s^5}}$

Step 1: Simplify the outer division:

This can be written as: $\frac{10t^3u}{3rs^3} \times \frac{9s^5}{5t^4u^4}$

Step 2: Simplify constants and variables:

• Constants: Simplify $\frac{10}{3} \times \frac{9}{5}$: $\frac{10}{3} \times \frac{9}{5} = \frac{90}{15} = 6$

• t terms: Simplify $t^3 \times t^{-4} = t^{-1} = \frac{1}{t}$

• u terms: Simplify $u \times u^{-4} = u^{-3} = \frac{1}{u^3}$

• s terms: Simplify $s^{-3} \times s^5 = s^2$

• r remains: The $r$ term stays in the denominator.

Step 3: Combine everything:

$\frac{6s^2}{rtu^3}$

Thus, the simplified expression is: $\boxed{\frac{6s^2}{rtu^3}}$

Would you like further details or have any other questions?

Here are 5 related questions for further practice:

1. How do you simplify an expression with multiple fractions?
2. What happens when you multiply powers with the same base?
3. How do you handle negative exponents in simplification?
4. Can you simplify expressions involving multiple variables more efficiently?
5. How do you combine like terms when simplifying algebraic expressions?

Tip: Always break down complex expressions step-by-step to avoid mistakes, especially when dealing with multiple variables and exponents.