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 exponents

Answer 1

To find the numerical equivalent of \( 12^{-5} \times 12^{7} \), we can apply the product rule of integer exponents. The product rule states that when multiplying two exponents with the same base, you add the exponents. In mathematical terms, this is represented as:

\[
a^m \times a^n = a^{m+n}
\]

In our specific case:

- The base \( a \) is \( 12 \).
- The exponent \( m \) is \( -5 \).
- The exponent \( n \) is \( 7 \).

Applying the product rule:

\[
12^{-5} \times 12^7 = 12^{-5 + 7}
\]

Now, we can calculate the new exponent:

\[
-5 + 7 = 2
\]

So we have:

\[
12^{-5} \times 12^7 = 12^2
\]

Now we need to find the numerical value of \( 12^2 \):

\[
12^2 = 12 \times 12 = 144
\]

Therefore, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is \( \boxed{144} \).

### Explanation of the Product Rule:
The product rule of exponents simplifies the process of multiplying exponents with the same base by allowing us to combine them into a single exponent. In this case, we added the exponents (–5 and 7), which resulted in a new exponent of \( 2 \). This simplifies our calculations significantly, especially when dealing with larger exponents.