https://file.aiquickdraw.com/mathbot/file/1727721618931rxhxma5h.jpg

Let's break down each expression in the image and compare them step by step.

We have three expressions to compare. These are:

1. \( \frac{3^4 \times 3^3}{12^3} \)
2. \( \frac{3^3 \times 4^3}{12^4} \)
3. \( \frac{3^4 \times 4^3}{12^3} \)

### Simplifying each expression:

#### First expression:
\[
\frac{3^4 \times 3^3}{12^3} = \frac{3^{4+3}}{12^3} = \frac{3^7}{12^3}
\]

#### Second expression:
\[
\frac{3^3 \times 4^3}{12^4} = \frac{3^3 \times 4^3}{(3 \times 4)^4} = \frac{3^3 \times 4^3}{3^4 \times 4^4} = \frac{3^3}{3^4} \times \frac{4^3}{4^4} = \frac{1}{3 \times 4} = \frac{1}{12}
\]

#### Third expression:
\[
\frac{3^4 \times 4^3}{12^3} = \frac{3^4 \times 4^3}{(3 \times 4)^3} = \frac{3^4 \times 4^3}{3^3 \times 4^3} = 3^{4-3} = 3
\]

### Now we can compare the three simplified results:
1. \( \frac{3^7}{12^3} \)
2. \( \frac{1}{12} \)
3. \( 3 \)

#### Comparison:
- \( \frac{3^7}{12^3} \) is a very large number.
- \( \frac{1}{12} \) is smaller than 1.
- \( 3 \) is clearly larger than \( \frac{1}{12} \), but smaller than \( \frac{3^7}{12^3} \).

Thus, the correct ordering of the expressions is:
\[
\frac{3^7}{12^3} > 3 > \frac{1}{12}
\]

Do you want more details or have any questions? Here are some related questions to explore further:

1. How can we generalize this comparison technique for any base and exponent combination?
2. What is the importance of simplifying expressions before comparing them?
3. What role do negative exponents play in simplifying fractions?
4. How do different base powers influence the growth of exponential expressions?
5. How does this problem help in understanding the properties of exponents?

**Tip:** Always simplify exponentiated terms before comparing them directly, as the results may be more intuitive!

Order the expressions by choosing >, <, or = for: 3^4 × 3^3 / 12^3, 3^3 × 4^3 / 12^4, and 3^4 × 4^3 / 12^3.

Learn how to compare exponential expressions with fractions like 3^4 × 3^3 / 12^3, 3^3 × 4^3 / 12^4, and 3^4 × 4^3 / 12^3. This problem involves key concepts of exponentiation, multiplication of powers, and simplification of fractions, suitable for students in Grades 9-11.

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