Apply the product rule of integer exponents to find the numerical equivalent of 12^-5 * 12^7. Show your work, and explain how do use the product rule of integer exponents

Answer 1

To find the numerical equivalent of \( 12^{-5} \times 12^{7} \), we can use the product rule of integer exponents, which states that when multiplying two powers with the same base, we add their exponents. The rule can be expressed as:

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

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

Applying the product rule, we have:

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

Now, we calculate the exponent:

\[ -5 + 7 = 2 \]

Thus, we get:

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

Next, we can find the numerical equivalent of \( 12^{2} \):

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

Therefore, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is \( 144 \).

In summary, we applied the product rule of integer exponents by adding the exponents of the same base, simplified to find \( 12^{2} \), and calculated the final numerical value as \( 144 \).

Answer 2

Can you do it a little shorter

Answer 3

Sure! Here’s a more concise explanation:

To find \( 12^{-5} \times 12^{7} \) using the product rule of integer exponents, we apply the rule:

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

Here, \( a = 12 \), \( m = -5 \), and \( n = 7 \):

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

Now, calculating \( 12^{2} \):

\[ 12^{2} = 144 \]

Thus, \( 12^{-5} \times 12^{7} = 144 \).