The given expression is:

\[
-4x^2 y^5 (-4x^5 y^5)
\]

To simplify this, follow these steps:

### Step 1: Multiply the constants
We have two constants, \(-4\) and \(-4\). Multiply them together:

\[
-4 \times -4 = 16
\]

### Step 2: Apply the product rule for exponents
The product rule states that when multiplying two terms with the same base, you add the exponents. Apply this to both \(x\) and \(y\) terms:

For \(x\)-terms: 
\[
x^2 \times x^5 = x^{2+5} = x^7
\]

For \(y\)-terms:
\[
y^5 \times y^5 = y^{5+5} = y^{10}
\]

### Step 3: Combine everything
Now combine all the simplified parts:

\[
16x^7 y^{10}
\]

Thus, the simplified form of the expression is:

\[
16x^7 y^{10}
\]

Would you like more details or have any questions?

### Related questions:
1. How do you simplify more complex algebraic expressions?
2. What are the rules for multiplying powers with the same base?
3. Can you apply the product rule for exponents in division as well?
4. What happens when the bases are different but the exponents are the same?
5. How would the result change if the terms had negative exponents?

**Tip**: Always group similar bases and apply exponent rules carefully to avoid mistakes in simplification!

Learn how to simplify the algebraic expression −4x^2 y^5 (−4x^5 y^5) using the product rule for exponents. This solution covers key concepts in exponentiation and algebra, including multiplying constants and applying exponent rules. Suitable for students in Grades 8-10.

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