https://file.aiquickdraw.com/mathbot/file/1729316763081ybed53u0.jpg

The equation provided in the image is:

\[
(27)^{5/2} \times (9)^{3/2} \div (3)^7 = (3)^x
\]

### Steps to Solve:

1. **Rewrite the bases in terms of powers of 3:**
   - \( 27 = 3^3 \), so \( (27)^{5/2} = (3^3)^{5/2} = 3^{(3 \cdot 5/2)} = 3^{15/2} \)
   - \( 9 = 3^2 \), so \( (9)^{3/2} = (3^2)^{3/2} = 3^{(2 \cdot 3/2)} = 3^3 \)

2. **Rewrite the equation using these powers:**

\[
3^{15/2} \times 3^3 \div 3^7 = 3^x
\]

3. **Combine exponents using the properties of exponents:**
   - Multiplying: \( 3^{15/2} \times 3^3 = 3^{(15/2 + 3)} = 3^{(15/2 + 6/2)} = 3^{21/2} \)
   - Dividing: \( 3^{21/2} \div 3^7 = 3^{(21/2 - 7)} = 3^{(21/2 - 14/2)} = 3^{7/2} \)

4. **Equating the powers of 3:**

\[
3^{7/2} = 3^x
\]

Since the bases are the same, the exponents must be equal:

\[
x = \frac{7}{2}
\]

### Final Answer:
\[
x = \frac{7}{2}
\]

Would you like any further details on this solution, or do you have additional questions?

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Here are 5 related questions:
1. How do you convert roots to fractional exponents?
2. What are the properties of exponents for multiplication and division?
3. How would you handle an equation like this if the bases were different?
4. Can this type of equation be solved using logarithms?
5. What are the common mistakes when simplifying powers of numbers?

**Tip:** When working with powers and roots, always try to express terms with a common base to simplify calculations easily.

Solve the equation: (27)^(5/2) × (9)^(3/2) ÷ (3)^7 = (3)^x

Learn how to solve an equation involving exponents: (27)^(5/2) × (9)^(3/2) ÷ (3)^7 = (3)^x. This problem covers key exponentiation concepts and algebraic properties. Suitable for high school students in Grades 9-12.

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