We are given the expression:

$\frac{5^2 \times 25 \times 3^3}{15 \times 9}$

Step 1: Simplify the powers and terms.

We know:

• $5^2 = 25$
• $25 = 5^2$
• $3^3 = 27$
• $15 = 3 \times 5$
• $9 = 3^2$

Thus, the expression becomes:

$\frac{5^2 \times 5^2 \times 3^3}{(3 \times 5) \times 3^2}$

Step 2: Combine like terms.

• In the numerator: $5^2 \times 5^2 = 5^{2+2} = 5^4$, so we get $5^4 \times 3^3$.
• In the denominator: $(3 \times 5) \times 3^2 = 3^{1+2} \times 5 = 3^3 \times 5$.

Now, the expression is:

$\frac{5^4 \times 3^3}{5 \times 3^3}$

Step 3: Cancel out common terms.

• $3^3$ in the numerator and denominator cancel out.
• $5^4$ and $5$ in the numerator and denominator cancel out, leaving $5^{4-1} = 5^3$.

Thus, the simplified expression is:

$5^3$

Final Answer:

The expression simplifies to $5^3$.

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Would you like a detailed explanation of any steps or help with another problem? Here are some related questions:

1. How do you handle exponents when multiplying powers with the same base?
2. What happens when you divide terms with the same base and powers?
3. How do you express fractions in exponential form?
4. What is the general rule for combining powers in expressions?
5. Can you simplify expressions with higher powers in the numerator and denominator?

Tip: When simplifying expressions, always look for common bases to combine terms and cancel out whenever possible.