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Let's go through the expressions and determine which are equivalent to \( 3^x \):

1. **A. \( \left( \frac{18}{6} \right)^x \):**
   Simplifying \( \frac{18}{6} = 3 \), so this becomes \( 3^x \), which **is** equivalent to the original expression.

2. **B. \( \frac{18^x}{3} \):**
   This is different. This expression does not simplify to \( 3^x \). Instead, it's \( \frac{18^x}{3} \), which is not equivalent to \( 3^x \). So, this is **not** equivalent.

3. **C. \( \frac{18^x}{6} \):**
   This also does not simplify to \( 3^x \). Like in B, this is not equivalent to \( 3^x \), as it represents \( \frac{18^x}{6} \). So, this is **not** equivalent.

4. **D. \( 3 \cdot 3^{x-1} \):**
   Using the properties of exponents, we know \( 3 \cdot 3^{x-1} = 3^1 \cdot 3^{x-1} = 3^{x} \). So, this is equivalent to \( 3^x \), meaning it **is** correct.

5. **E. \( 3 \cdot 3^{x+1} \):**
   This simplifies to \( 3^1 \cdot 3^{x+1} = 3^{x+2} \), which is not the same as \( 3^x \). So, this is **not** equivalent.

6. **F. \( x^3 \):**
   This expression is clearly different from \( 3^x \). It represents \( x \) raised to the power of 3, not 3 raised to the power of \( x \). Therefore, it is **not** equivalent.

### Correct answers: A and D

Would you like more details or have any questions? 

Here are five related questions to deepen understanding:
1. How do exponent rules apply when multiplying terms with the same base?
2. What happens when dividing exponents with the same base?
3. Can you explain why \( \frac{18^x}{3} \) and \( 3^x \) are not equivalent?
4. How does adding or subtracting exponents affect the expression?
5. In what situations is \( x^3 \) equivalent to \( 3^x \), if ever?

**Tip**: When dealing with exponents, always apply the basic properties of exponents (e.g., \( a^m \cdot a^n = a^{m+n} \)) to simplify expressions effectively.

Which expressions are equivalent to the one below? Check all that apply.
3^x

Learn which expressions are equivalent to 3^x by applying exponent rules. This problem focuses on exponentiation, simplification, and the properties of exponents. Suitable for students in Grades 9-11.

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