Let's carefully analyze the problem step-by-step to determine if Charlie's work is correct and which of the given statements is true.
1. Understanding the problem:
- We need to verify Charlie's calculation for [tex]\( 3^3 \cdot 3^2 \)[/tex].
2. Applying the rules of exponents:
- We know that when multiplying powers with the same base, we add the exponents. Specifically, [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]. Here, the base [tex]\( a \)[/tex] is 3, [tex]\( m \)[/tex] is 3, and [tex]\( n \)[/tex] is 2.
3. Performing the exponent addition:
- According to the rule above:
[tex]\[
3^3 \cdot 3^2 = 3^{3+2}
\][/tex]
- Simplifying the exponents:
[tex]\[
3^{3+2} = 3^5
\][/tex]
4. Calculating the actual value:
- We calculate [tex]\( 3^5 \)[/tex]:
[tex]\[
3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243
\][/tex]
5. Evaluating Charlie's steps:
- Charlie states:
[tex]\[
3^3 \cdot 3^2 = 3^6
\][/tex]
- But we know from the correct calculation that it should be [tex]\( 3^5 \)[/tex], not [tex]\( 3^6 \)[/tex].
6. Calculating Charlie's result:
- Charlie incorrectly calculated:
[tex]\[
3^6 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 729
\][/tex]
- This is incorrect because the exponent should have been 5, not 6.
7. Assessing the given statements:
- Let's review each statement:
- Statement 1: "He is incorrect because he should have 3 factors of 6." This statement is not relevant to the scenario.
- Statement 2: "He is correct." This statement is false because Charlie's calculation of [tex]\( 3^6 \)[/tex] is incorrect.
- Statement 3: "He is incorrect because he should have 6 factors of 9." This statement is neither correct nor relevant.
- Statement 4: "He is incorrect because he should have only 5 factors of 3." This statement is true. Correctly, [tex]\( 3^3 \cdot 3^2 \)[/tex] simplifies to [tex]\( 3^5 \)[/tex], which means there should be 5 factors of 3.
Thus, the correct statement is:
"He is incorrect because he should have only 5 factors of 3."
