https://file.aiquickdraw.com/mathbot/file/1726207145034afx1t5nz.jpg

The given expression is:

\[
\frac{3^2 \times 10^5 \times 25}{5^3 \times 6^4}
\]

Let's simplify this step by step.

### Step 1: Break down the numbers
- \( 3^2 = 9 \)
- \( 10^5 \) remains as is.
- \( 25 = 5^2 \), so \( 25 \) can be rewritten as \( 5^2 \).
- \( 5^3 \) remains as is.
- \( 6^4 = (2 \times 3)^4 = 2^4 \times 3^4 = 16 \times 81 = 1296 \).

Now, we rewrite the expression as:

\[
\frac{9 \times 10^5 \times 5^2}{5^3 \times 1296}
\]

### Step 2: Simplify the powers of 5
The numerator has \( 5^2 \) and the denominator has \( 5^3 \). Using the rule \( \frac{a^m}{a^n} = a^{m-n} \), we get:

\[
5^{2-3} = 5^{-1} = \frac{1}{5}
\]

So the expression becomes:

\[
\frac{9 \times 10^5}{5 \times 1296}
\]

### Step 3: Simplify the constants
Now, simplify the constants:

\[
\frac{9}{5 \times 1296} = \frac{9}{6480} = \frac{3}{2160} = \frac{1}{720}
\]

Thus, the simplified expression is:

\[
\frac{10^5}{720}
\]

### Final Answer:
The simplified form of the expression is:

\[
\frac{10^5}{720}
\]

Would you like more details or have further questions?

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Here are 5 related questions:
1. How do you simplify powers when multiplying or dividing them?
2. What are the properties of exponents used in this simplification?
3. How do you handle fractions with large numbers like \( \frac{10^5}{720} \)?
4. What are the common factorizations for numbers like 1296?
5. How does the rule \( \frac{a^m}{a^n} = a^{m-n} \) help in simplifying expressions?

**Tip:** Always break down composite numbers into their prime factors when simplifying expressions involving exponents.

Simplify the expression: (3^2 * 10^5 * 25) / (5^3 * 6^4)

Learn how to simplify the exponent expression (3^2 * 10^5 * 25) / (5^3 * 6^4) using the laws of exponents. This detailed step-by-step solution covers key mathematical concepts such as multiplication of powers and prime factorization. Suitable for students in Grades 9-10.

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