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Simplify the expression using the​ products-to-powers rule.

To simplify the given expression \( (-5y^2)^4 \), we will apply the **power of a product rule**, which states that:

\[
(a \cdot b)^n = a^n \cdot b^n
\]

In this case, the product involves \(-5\) and \(y^2\), and the exponent is 4. Now let's break it down:

1. Apply the exponent to \(-5\):
   \[
   (-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 625
   \]

2. Apply the exponent to \(y^2\):
   \[
   (y^2)^4 = y^{2 \times 4} = y^8
   \]

Now multiply the two results:
\[
625 \cdot y^8 = 625y^8
\]

Thus, the simplified expression is:
\[
625y^8
\]

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you simplify \( (-3x^3)^5 \)?
2. What is the result of \( (2xy^2)^3 \)?
3. How does the power of a quotient rule differ from the power of a product rule?
4. How would you simplify an expression like \( (-7z^4)^3 \)?
5. What happens when an expression like \( (-2x)^0 \) is simplified?

**Tip:** Always remember that raising a negative number to an even exponent results in a positive number, but raising it to an odd exponent keeps the result negative.

Simplify the expression using the products-to-powers rule: (-5y^2)^4

Learn how to simplify the expression (-5y^2)^4 by applying the products-to-powers rule. This solution includes step-by-step instructions to break down the exponent for both constants and variables. Suitable for Grades 9-12 students.

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