https://file.aiquickdraw.com/mathbot/file/1728509064508jmtv2r6t.jpg

Let's go through the simplification of the expression provided in the image step by step:

The given expression is:

\[
(5x^2y^3)^2 \cdot (-3xy^3)^3
\]

### Step 1: Apply the powers to each term
For both parts of the expression, distribute the exponents to each term inside the parentheses.

#### First part: \((5x^2y^3)^2\)
- Apply the exponent 2 to each factor inside the parentheses:
  \[
  (5^2)(x^{2 \cdot 2})(y^{3 \cdot 2}) = 25x^4y^6
  \]

#### Second part: \((-3xy^3)^3\)
- Apply the exponent 3 to each factor inside the parentheses:
  \[
  (-3^3)(x^{1 \cdot 3})(y^{3 \cdot 3}) = -27x^3y^9
  \]

### Step 2: Multiply the two expressions
Now, multiply the results from Step 1:
\[
25x^4y^6 \cdot (-27x^3y^9)
\]

- Multiply the constants:
  \[
  25 \cdot (-27) = -675
  \]

- Multiply the powers of \(x\):
  \[
  x^4 \cdot x^3 = x^{4 + 3} = x^7
  \]

- Multiply the powers of \(y\):
  \[
  y^6 \cdot y^9 = y^{6 + 9} = y^{15}
  \]

### Final simplified expression:
\[
-675x^7y^{15}
\]

So, the answer is:

\[
\boxed{-675x^7y^{15}}
\]

Do you want more details on any step? Here are five questions related to this explanation:

1. How do you apply the power of a product rule to simplify terms like \((5x^2y^3)^2\)?
2. What is the process for multiplying terms with the same base in algebraic expressions?
3. Why does \((-3xy^3)^3\) result in a negative number after simplification?
4. Can we generalize the power rule for any algebraic expression of the form \((a^m b^n)^p\)?
5. How do you handle negative bases when raised to odd exponents?

**Tip:** Always remember to carefully distribute exponents to all parts inside parentheses, especially when dealing with negative signs.

Learn how to simplify the expression (5x^2y^3)^2(-3xy^3)^3 step by step. This problem covers key algebraic concepts such as the power of a product rule and simplifying expressions with exponents. Suitable for students in Grades 9-11.

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