Question

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

Answer 1

To solve the expression \( 12^{-5} \cdot 12^{7} \), we will apply the product rule of integer exponents, which states that when multiplying two expressions with the same base, you can add their exponents. The rule can be written as:

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

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

Using the product rule, we can combine the exponents:

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

Now, we perform the addition of the exponents:

\[
-5 + 7 = 2
\]

Thus, we have:

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

Next, we can calculate \( 12^{2} \):

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

So, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:

\[
\boxed{144}
\]

In summary, we applied the product rule of integer exponents to combine the exponents of the same base and calculated the resulting exponent to find the final numerical value.