Question

Which expression is correctly developed to use the Power Rule of Exponents?

1. (7 to the fourth power) to the eight power?
2. 12 to the ninth power?
3. (7 to the eighth power) times (7 to the tenth power)?
4. (-5) to the zero power?

Answer 1

To determine which expression is correctly developed to use the Power Rule of Exponents, we need to assess each option based on the Power Rule, which states that \( (a^m)^n = a^{m \cdot n} \).

1. **(7 to the fourth power) to the eighth power**:
- This can be expressed as \( (7^4)^8 = 7^{4 \cdot 8} = 7^{32} \).
- This expression correctly applies the Power Rule.

2. **12 to the ninth power**:
- This is simply \( 12^9 \).
- This doesn't invoke the Power Rule as it stands as a single exponent without further manipulation.

3. **(7 to the eighth power) times (7 to the tenth power)**:
- This can be simplified using the Product Rule of Exponents, which states that \( a^m \cdot a^n = a^{m+n} \).
- So, it becomes \( 7^8 \cdot 7^{10} = 7^{8 + 10} = 7^{18} \).
- This option uses the Product Rule, not the Power Rule.

4. **(-5) to the zero power**:
- According to the Zero Exponent Rule, any non-zero number raised to the zero power is 1: \( (-5)^0 = 1 \).
- This expression does not include the Power Rule.

Based on this analysis, **option 1** is the correctly developed expression that uses the Power Rule of Exponents.