To simplify the given expression, let's work step by step:

The expression is:

$\frac{3r^{-1}st}{3r^{-6}s^{3}t}.$

Step 1: Cancel the coefficients

The coefficient "3" is the same in the numerator and denominator, so it cancels out:

$\frac{r^{-1}st}{r^{-6}s^{3}t}.$

Step 2: Simplify the variables $r$, $s$, and $t$

$r$ terms:

Use the rule of exponents $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{r^{-1}}{r^{-6}} = r^{-1 - (-6)} = r^{-1 + 6} = r^5.$

$s$ terms:

$\frac{s}{s^3} = s^{1-3} = s^{-2}.$

$t$ terms:

$\frac{t}{t} = t^{1-1} = t^0 = 1.$

Step 3: Combine the results

Now the expression simplifies to:

$r^5s^{-2}.$

Step 4: Write with positive exponents

To express $s^{-2}$ with a positive exponent, rewrite it as $\frac{1}{s^2}$:

$r^5s^{-2} = \frac{r^5}{s^2}.$

Final Answer:

$\boxed{\frac{r^5}{s^2}}$

Would you like additional details or examples on exponent rules?

Here are 5 related questions to expand your understanding:

1. How would you simplify $\frac{x^{-3}y^2}{x^{-5}y^{-4}}$ using positive exponents?
2. What are the general rules for simplifying expressions with negative exponents?
3. If $a^m \cdot a^n = a^{m+n}$, how would this apply to division?
4. How do you simplify expressions with coefficients, such as $\frac{6x^{-2}}{3x^4}$?
5. What does $x^0 = 1$ represent, and why does it hold for any nonzero $x$?

Tip: Always handle coefficients and variables separately when simplifying algebraic fractions.