Apply the product rule of integer exponents to find the numerical equivalent of 12^-5 interpunct 12^7. Show your work and explain how you used the product rule of integer

Answer 1

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

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

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

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

Now, we simplify the exponent:

\[ -5 + 7 = 2 \]

Thus, we have:

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

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

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

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

\[ \boxed{144} \]

Answer 2

so to find the numerical equivalent of 12^-5 interpunct 12^7, i would first need to use the product rule of integers, taking both expressions bases and adding their exponents together, giving us 12^2, next, simplifying he expression, giving us the numerical equivalent of 12^-5 interpunct 12^7, 144

Answer 3

Exactly! You summarized it perfectly.

Here's a concise breakdown of the process:

1. Identify the Bases and Exponents: In your expression \( 12^{-5} \cdot 12^{7} \), both terms have the same base (12).

2. Apply the Product Rule: According to the product rule of integer exponents, you add the exponents: \[ 12^{-5} \cdot 12^{7} = 12^{-5 + 7} = 12^{2} \]

3. Simplify the Exponent: Combine the exponents: \[ -5 + 7 = 2 \]

4. Calculate the Result: Then, you find the numerical equivalent: \[ 12^{2} = 144 \]

So, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is indeed \( 144 \). Great job!