Question

Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127
12
−
5
⋅
12
7
. Show your work and explain how you used the product rule of integer exponents.(2 points)
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Answer 1

To find the numerical equivalent of \( 12^{-5} \cdot 12^7 \), we can apply the product rule of integer exponents, which states that when multiplying two powers with the same base, we can add their exponents. That is:

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

In our case, the base \( a \) is \( 12 \), \( m = -5 \), and \( n = 7 \).

Now we can apply the product rule:

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

Next, we calculate the exponent:

\[
-5 + 7 = 2
\]

So, we can rewrite the expression as:

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

Now we can calculate \( 12^2 \):

\[
12^2 = 144
\]

Thus, the numerical equivalent of \( 12^{-5} \cdot 12^7 \) is \( 144 \).

In summary:

1. We applied the product rule of exponents to combine the terms: \( 12^{-5} \cdot 12^7 = 12^{-5 + 7} = 12^{2} \).
2. We evaluated \( 12^{2} \) to get \( 144 \).

Final answer: \( 144 \)