Question

An expression was simplified to 24 ^ 12 Which expression correctly used the Product Rule of Exponents to simplify?

(1 point)

Ο 24 ^ 3 * 24 ^ 4

24 ^ 4 - 24 ^ 4

6 ^ 4 * 4 ^ 3

12 ^ 4 * 12 ^ 4

Answer 1

To find the expression that correctly used the Product Rule of Exponents to simplify to \( 24^{12} \), we should apply the Product Rule, which states that when multiplying two powers with the same base, you add their exponents:

\[
a^m \cdot a^n = a^{m+n}
\]

Now, let's analyze the provided options:

1. **\( 24^{3} \cdot 24^{4} \)**:
- According to the Product Rule:
\[
24^{3} \cdot 24^{4} = 24^{3+4} = 24^{7}
\]
This does not simplify to \( 24^{12} \).

2. **\( 24^{4} - 24^{4} \)**:
- This simplifies to zero:
\[
24^{4} - 24^{4} = 0
\]
This is clearly not \( 24^{12} \).

3. **\( 6^{4} \cdot 4^{3} \)**:
- To see if this could be equivalent to \( 24^{12} \), we can rewrite \( 24 \):
\[
24 = 6 \cdot 4
\]
But multiplying these does not directly apply to the powers correctly to yield \( 24^{12} \).

4. **\( 12^{4} \cdot 12^{4} \)**:
- This simplifies as follows using the Product Rule:
\[
12^{4} \cdot 12^{4} = 12^{4+4} = 12^{8}
\]
This does not yield \( 24^{12} \).

None of the options provided correctly simplifies to \( 24^{12} \) using the Product Rule. However, just for clarity and correction, if you had instead \( 12^{6} \cdot 12^{6} \):
\[
12^{6} \cdot 12^{6} = 12^{6+6} = 12^{12}
\]
And if \( 12 \) could be expressed in terms of \( 24 \) equivalently, then that would need consideration.

Thus, none of the expressions listed yield \( 24^{12} \) using the Product Rule directly. If you have any additional options or follow-up questions, please provide them!