Question
Apply the product rule of integer exponents to find the numerical equivalent of 12-5 x 12^7. Show your work and explain how you used the product rule of integer exponents. (in a 8th graders words
Answers
GPT-4o mini
To solve the expression \( 12^{-5} \times 12^7 \), we can use the product rule of integer exponents. The product rule states that when you multiply two powers that have the same base, you can add their exponents. The rule looks like this:
\[
a^m \times a^n = a^{m+n}
\]
In our case, the base is \( 12 \), and our exponents are \( -5 \) and \( 7 \).
1. **Identify the base and exponents**: Here, our base is \( 12 \) and the exponents are \( -5 \) and \( 7 \).
2. **Add the exponents**: According to the product rule, we can add these exponents together:
\[
-5 + 7 = 2
\]
3. **Combine the base and the new exponent**: Now we can rewrite the expression using the result from the addition:
\[
12^{-5} \times 12^7 = 12^{2}
\]
4. **Calculate \( 12^{2} \)**: This means we need to find out what \( 12^2 \) equals:
\[
12 \times 12 = 144
\]
So, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is **144**.
In summary, we used the product rule of integer exponents to combine the two powers of \( 12 \) by adding their exponents together, which simplified our expression to \( 12^{2} \), and we found that \( 12^2 = 144 \).
\[
a^m \times a^n = a^{m+n}
\]
In our case, the base is \( 12 \), and our exponents are \( -5 \) and \( 7 \).
1. **Identify the base and exponents**: Here, our base is \( 12 \) and the exponents are \( -5 \) and \( 7 \).
2. **Add the exponents**: According to the product rule, we can add these exponents together:
\[
-5 + 7 = 2
\]
3. **Combine the base and the new exponent**: Now we can rewrite the expression using the result from the addition:
\[
12^{-5} \times 12^7 = 12^{2}
\]
4. **Calculate \( 12^{2} \)**: This means we need to find out what \( 12^2 \) equals:
\[
12 \times 12 = 144
\]
So, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is **144**.
In summary, we used the product rule of integer exponents to combine the two powers of \( 12 \) by adding their exponents together, which simplified our expression to \( 12^{2} \), and we found that \( 12^2 = 144 \).