Let's simplify the expression step by step:

$\frac{(-9x^4y^7)^2}{-27x^{10}y^{16}}$

Step 1: Expand the numerator

We begin by squaring the terms in the numerator:

$(-9x^4y^7)^2 = (-9)^2(x^4)^2(y^7)^2 = 81x^8y^{14}$

So the expression becomes:

$\frac{81x^8y^{14}}{-27x^{10}y^{16}}$

Step 2: Simplify the coefficients

Now, simplify the numerical coefficients:

$\frac{81}{-27} = -3$

Step 3: Simplify the powers of $x$

Use the quotient rule for exponents $\frac{x^m}{x^n} = x^{m-n}$:

$\frac{x^8}{x^{10}} = x^{8-10} = x^{-2}$

Step 4: Simplify the powers of $y$

Similarly, for $y$:

$\frac{y^{14}}{y^{16}} = y^{14-16} = y^{-2}$

Step 5: Final expression

Combining all terms:

$-3x^{-2}y^{-2} = \frac{-3}{x^2y^2}$

Thus, the simplified expression is:

$\frac{-3}{x^2y^2}$

Would you like further details or have any questions?

Here are five related questions:

1. How do you simplify powers with negative exponents?
2. What is the rule for simplifying fractions with the same base?
3. Can you square both negative and positive numbers?
4. How do you handle negative exponents in fractions?
5. What is the importance of reducing coefficients in algebraic expressions?

Tip: Always apply exponent rules carefully when simplifying algebraic expressions.