The provided question asks to simplify the following expression:

$\frac{3m^2n^{-3}}{15n^2m^4}$

Step 1: Simplify the constants

The constants in the expression are 3 and 15. Simplify this as:

$\frac{3}{15} = \frac{1}{5}$

Step 2: Simplify the $m$-terms

You have $m^2$ in the numerator and $m^4$ in the denominator. Using the law of exponents $\frac{a^m}{a^n} = a^{m-n}$, we subtract the exponents of $m$:

$\frac{m^2}{m^4} = m^{2-4} = m^{-2}$

Step 3: Simplify the $n$-terms

You have $n^{-3}$ in the numerator and $n^2$ in the denominator. Similarly, subtract the exponents of $n$:

$\frac{n^{-3}}{n^2} = n^{-3-2} = n^{-5}$

Step 4: Combine everything

Now, combining all the simplified terms together:

$\frac{3m^2n^{-3}}{15n^2m^4} = \frac{m^{-2}n^{-5}}{5}$

Thus, the simplified expression is:

$\frac{m^{-2}n^{-5}}{5}$

Step 5: Match with the answer choices

This matches option A:

$\boxed{A. \frac{m^{-2}n^{-5}}{5}}$

Let me know if you need more details or have any questions.

Here are 5 related questions for further practice:

1. Simplify $\frac{6a^3b^{-2}}{18a^5b}$.
2. Simplify $\frac{4x^2y^{-4}}{16x^{-3}y^5}$.
3. Find the result of $\frac{5p^4q^{-1}}{20p^2q^3}$.
4. Simplify $\frac{7x^{-2}y^3}{14x^5y^{-2}}$.
5. Solve $\frac{2m^{-1}n^2}{8m^3n^{-1}}$.

Tip:

When simplifying algebraic fractions, always handle constants separately and apply exponent rules like $\frac{a^m}{a^n} = a^{m-n}$ for like bases.