Question
Which expression is correctly developed to use the Product Rule of Exponents?
10 to the eighth power times 10 to the eighth power
(5 to the second power) to the ninth power
32 to the seventh power
6 to the second power times 7 to the third power
10 to the eighth power times 10 to the eighth power
(5 to the second power) to the ninth power
32 to the seventh power
6 to the second power times 7 to the third power
Answers
GPT-4o mini
The Product Rule of Exponents states that when you multiply two expressions with the same base, you add their exponents.
Let's analyze each expression:
1. **\(10^8 \times 10^8\)**: According to the Product Rule, this can be simplified to \(10^{8 + 8} = 10^{16}\).
2. **\((5^2)^9\)**: This expression uses the Power Rule of Exponents, which states that when you raise a power to another power, you multiply the exponents. So, this becomes \(5^{2 \times 9} = 5^{18}\), rather than using the Product Rule.
3. **\(32^7\)**: This expression does not involve multiplication of two identical bases, so it does not apply the Product Rule.
4. **\(6^2 \times 7^3\)**: This involves two different bases (6 and 7). The Product Rule does not apply here since it requires the same base for the multiplication.
From the analysis, the only expression that correctly represents the application of the Product Rule is:
- **\(10^8 \times 10^8\)**, which simplifies to \(10^{16}\).
Let's analyze each expression:
1. **\(10^8 \times 10^8\)**: According to the Product Rule, this can be simplified to \(10^{8 + 8} = 10^{16}\).
2. **\((5^2)^9\)**: This expression uses the Power Rule of Exponents, which states that when you raise a power to another power, you multiply the exponents. So, this becomes \(5^{2 \times 9} = 5^{18}\), rather than using the Product Rule.
3. **\(32^7\)**: This expression does not involve multiplication of two identical bases, so it does not apply the Product Rule.
4. **\(6^2 \times 7^3\)**: This involves two different bases (6 and 7). The Product Rule does not apply here since it requires the same base for the multiplication.
From the analysis, the only expression that correctly represents the application of the Product Rule is:
- **\(10^8 \times 10^8\)**, which simplifies to \(10^{16}\).