Question

An expression was simplified to 24 to the twelfth power. Which expression correctly used the Product Rule of Exponents to simplify?

12 to the fourth power times 12 to the eighth power
6 to the fourth power times 4 to the third power
24 to the third power times 24 to the fourth power
24 to the fourth power times 24 to the eighth power

Answer 1

To determine which expression correctly simplifies to \( 24^{12} \) using the Product Rule of Exponents, we apply the Product Rule, which states that \( a^m \times a^n = a^{m+n} \).

Let's analyze each expression:

1. **\( 12^4 \times 12^8 \)**:
\[
12^4 \times 12^8 = 12^{4+8} = 12^{12} \quad \text{(Not equal to \( 24^{12} \))}
\]

2. **\( 6^4 \times 4^3 \)**:
\[
6^4 \times 4^3 = (6^4) \times (4^3) \quad \text{(Cannot be simplified to \( 24^{12} \))}
\]

3. **\( 24^3 \times 24^4 \)**:
\[
24^3 \times 24^4 = 24^{3+4} = 24^7 \quad \text{(Not equal to \( 24^{12} \))}
\]

4. **\( 24^4 \times 24^8 \)**:
\[
24^4 \times 24^8 = 24^{4+8} = 24^{12}
\]

The expression that correctly simplifies to \( 24^{12} \) using the Product Rule of Exponents is:

**\( 24^4 \times 24^8 \)**.