https://file.aiquickdraw.com/mathbot/file/1728579596844axgbp5jo.jpg

The expression shown in the image is:

\[
\frac{3^7 + 5 \cdot 3^4}{6^5}
\]

### Step 1: Simplify the numerator
We first need to calculate the two terms in the numerator:

1. \( 3^7 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2187 \)
2. \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \), so \( 5 \cdot 3^4 = 5 \times 81 = 405 \)

Now add the results in the numerator:

\[
3^7 + 5 \cdot 3^4 = 2187 + 405 = 2592
\]

### Step 2: Calculate the denominator
Next, we calculate \( 6^5 \):

\[
6^5 = 6 \times 6 \times 6 \times 6 \times 6 = 7776
\]

### Step 3: Combine the numerator and denominator
Now divide the simplified numerator by the denominator:

\[
\frac{2592}{7776} = \frac{1}{3}
\]

### Final Answer:
\[
\frac{1}{3}
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand your understanding:

1. What are the general rules for simplifying exponents in an expression?
2. How can we use logarithms to simplify powers of large numbers?
3. What are some properties of exponents that help in reducing complex expressions?
4. How do you simplify expressions with different bases, like \( 3^a \) and \( 6^b \)?
5. Can you think of a real-world scenario where powers and exponents like these might be used?

**Tip:** Always simplify the powers first and handle multiplication before addition when solving these types of problems.

Solve the expression: (3^7 + 5 * 3^4) / 6^5

Learn how to simplify the expression (3^7 + 5 * 3^4) / 6^5 step-by-step, including calculations for exponents and combining terms. This problem covers exponentiation rules and is ideal for Grades 9-10 students.

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