Question

Apply the Product Rule of Integer Exponents to find an equivalent expression for and then solve the expression 12^−5⋅12^7
. You final answer should not include exponents.

This question is worth 2 points. You will earn one point for the correct answer, and one point for your work/explanation.

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(2 points)

Answer 1

To solve the expression \( 12^{-5} \cdot 12^7 \) using the Product Rule of Integer Exponents, we can apply the rule which states that when multiplying two powers with the same base, we add the exponents:

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

In this case, \( a = 12 \), \( m = -5 \), and \( n = 7 \). Applying the Product Rule:

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

Now, we need to simplify \( 12^{2} \):

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

Thus, the equivalent expression for \( 12^{-5} \cdot 12^7 \) is \( 12^2 \), which evaluates to:

\[
\boxed{144}
\]

So the final answer, without exponents, is \( 144 \).