Question

Is the expression x3 ∙ x3 ∙ x3 equivalent to x3 ∙ 3 ∙ 3? Why or why not? Explain your reasoning.

Answer 1

To determine if the expression \( x^3 \cdot x^3 \cdot x^3 \) is equivalent to \( x^3 \cdot 3 \cdot 3 \), we need to analyze each expression separately.

1. **Analyzing \( x^3 \cdot x^3 \cdot x^3 \):**
- This expression involves multiplying the term \( x^3 \) three times.
- By the properties of exponents, specifically the rule that states \( a^m \cdot a^n = a^{m+n} \), we can combine the powers:
\[
x^3 \cdot x^3 \cdot x^3 = x^{3+3+3} = x^9.
\]

2. **Analyzing \( x^3 \cdot 3 \cdot 3 \):**
- This expression consists of \( x^3 \) multiplied by 3 two times.
- Multiplying the constants first:
\[
3 \cdot 3 = 9.
\]
- Therefore, the expression simplifies to:
\[
x^3 \cdot 9.
\]

Now, we can summarize the two results:
- The first expression \( x^3 \cdot x^3 \cdot x^3 \) simplifies to \( x^9 \).
- The second expression \( x^3 \cdot 3 \cdot 3 \) simplifies to \( 9x^3 \).

Since \( x^9 \) is not equal to \( 9x^3 \), we conclude that the two expressions \( x^3 \cdot x^3 \cdot x^3 \) and \( x^3 \cdot 3 \cdot 3 \) are **not equivalent**.