Question
apply the product rule of integer exponents to find the numerical equivalent of 12 to the power of -5 x 12 to the power of 7 show your work and explain how you used the product rule of integer exponents . write it sounding 6th grade
Answers
GPT-4o mini
To solve \( 12^{-5} \times 12^{7} \), we can use the product rule of integer exponents. The product rule says that when you multiply two numbers with the same base (in this case, 12), you can add their exponents.
Here's how it works step by step:
1. **Identify the Base and Exponents**: The base here is \( 12 \), and the exponents are \( -5 \) and \( 7 \).
2. **Use the Product Rule**: According to the product rule \( a^m \times a^n = a^{m+n} \):
- Here, \( m = -5 \) and \( n = 7 \).
- So, we can add the exponents:
\[
-5 + 7 = 2
\]
3. **Rewrite the Expression**: Now we can rewrite the multiplication using the summed exponent:
\[
12^{-5} \times 12^{7} = 12^{2}
\]
4. **Calculate the Value**: Now we need to find the value of \( 12^{2} \):
\[
12^{2} = 12 \times 12 = 144
\]
So, \( 12^{-5} \times 12^{7} = 144 \).
**Summary**: We used the product rule of exponents to add the exponents together since both parts of our multiplication had the same base (12). That helped us simplify \( 12^{-5} \times 12^{7} \) to easily find the answer, which is \( 144 \).
Here's how it works step by step:
1. **Identify the Base and Exponents**: The base here is \( 12 \), and the exponents are \( -5 \) and \( 7 \).
2. **Use the Product Rule**: According to the product rule \( a^m \times a^n = a^{m+n} \):
- Here, \( m = -5 \) and \( n = 7 \).
- So, we can add the exponents:
\[
-5 + 7 = 2
\]
3. **Rewrite the Expression**: Now we can rewrite the multiplication using the summed exponent:
\[
12^{-5} \times 12^{7} = 12^{2}
\]
4. **Calculate the Value**: Now we need to find the value of \( 12^{2} \):
\[
12^{2} = 12 \times 12 = 144
\]
So, \( 12^{-5} \times 12^{7} = 144 \).
**Summary**: We used the product rule of exponents to add the exponents together since both parts of our multiplication had the same base (12). That helped us simplify \( 12^{-5} \times 12^{7} \) to easily find the answer, which is \( 144 \).