Question
Apply the product rule of integer exponents to find the numerical equivalent of 12 superscript −5⋅12 superscript 7
. Show your work and explain how you used the product rule of integer exponents.(2 points)
. Show your work and explain how you used the product rule of integer exponents.(2 points)
Answers
GPT-4o mini
To find the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) using the product rule of integer exponents, we start by recalling the product rule itself. The product rule states that when multiplying 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 \( 12 \), \( m = -5 \), and \( n = 7 \). We can apply the product rule as follows:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
Now we perform the addition in the exponent:
\[
-5 + 7 = 2
\]
So, we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
Next, we calculate \( 12^{2} \):
\[
12^{2} = 12 \cdot 12 = 144
\]
Thus, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:
\[
\boxed{144}
\]
In summary, by applying the product rule, we combined the exponents of the same base and simplified the expression to arrive at the final result of 144.
\[
a^m \cdot a^n = a^{m+n}
\]
In our case, the base \( a \) is \( 12 \), \( m = -5 \), and \( n = 7 \). We can apply the product rule as follows:
\[
12^{-5} \cdot 12^{7} = 12^{-5 + 7}
\]
Now we perform the addition in the exponent:
\[
-5 + 7 = 2
\]
So, we have:
\[
12^{-5} \cdot 12^{7} = 12^{2}
\]
Next, we calculate \( 12^{2} \):
\[
12^{2} = 12 \cdot 12 = 144
\]
Thus, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:
\[
\boxed{144}
\]
In summary, by applying the product rule, we combined the exponents of the same base and simplified the expression to arrive at the final result of 144.