Question

Apply the product rule of integer exponents to find the numerical equivalent of 12 to the power of -5 and 12 to the power of 7

Answers

Answered by GPT-4o mini
To solve \( 12^{-5} \) and \( 12^{7} \), we apply the product rule of integer exponents, which states that \( a^{m} \cdot a^{n} = a^{m+n} \).

We are interested in finding \( 12^{-5} \) and \( 12^{7} \):

1. **Calculate \( 12^{-5} \)**:
\[
12^{-5} = \frac{1}{12^{5}}
\]

2. **Calculate \( 12^{7} \)**:
\[
12^{7} = 12^{7}
\]

Next, we can simplify the expression using the product rule:

To combine these two powers, we would calculate:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7} = 12^{2}
\]

Now, we can find the numerical equivalent of \( 12^{2} \):
\[
12^{2} = 144
\]

Thus, the product of \( 12^{-5} \) and \( 12^{7} \) is:

\[
12^{-5} \cdot 12^{7} = 144
\]

And individually:
- \( 12^{-5} \) is equal to \( \frac{1}{12^{5}} \), which is a very small number.
- \( 12^{7} = 893871739 \) (if you want the numeric value as well).

So, the results are:
- \( 12^{-5} \) = \( \frac{1}{248832} \) (since \( 12^{5} = 248832 \))
- \( 12^{7} = 893871739 \)
- Product of \( 12^{-5} \) and \( 12^{7} \) = \( 144 \).
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