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

Answer 1

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

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

In our case, the base \( a \) is \( 12 \), \( m \) is \( -5 \), and \( n \) is \( 7 \). According to the product rule, we can combine the exponents:

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

Now, we will calculate the exponent:

\[ -5 + 7 = 2 \]

Thus, we have:

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

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

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

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

\[ \boxed{144} \]