To find the numerical equivalent of \( 12^{-5} \), we can use the product rule of integer exponents. The product rule states that when you multiply powers with the same base, you add the exponents. But first, let's rewrite \( 12^{-5} \).
When there's a negative exponent, it means we take the reciprocal of the base raised to the positive exponent. So:
\[ 12^{-5} = \frac{1}{12^5} \]
Now, we need to calculate \( 12^5 \). We can break this down like this:
\[ 12^5 = 12 \times 12 \times 12 \times 12 \times 12 \]
Calculating this step by step, we can find \( 12^2 \):
\[ 12^2 = 144 \]
Next, calculate \( 12^3 \):
\[ 12^3 = 12 \times 144 = 1728 \]
Now \( 12^4 \):
\[ 12^4 = 12 \times 1728 = 20736 \]
And finally:
\[ 12^5 = 12 \times 20736 = 248832 \]
Now, we can write \( 12^{-5} \):
\[ 12^{-5} = \frac{1}{248832} \]
So, the numerical equivalent of \( 12^{-5} \) is \( \frac{1}{248832} \).
In summary, we used the property of negative exponents to find the reciprocal and calculated \( 12^5 \) step by step to arrive at the answer.