Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.(2 points) make it sound like an eleven year old

Answer 1

Alright! So, let’s solve \( 12^{-5} \cdot 12^7 \), and I'll show you how I used the product rule of integer exponents!

First, the product rule of exponents says that when you're multiplying two numbers with the same base (like \( 12 \) in this case), you can just add the exponents. So, if we have:

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

For our example, we can think of it like this:

• The base is \( 12 \).
• The first exponent is \( -5 \) (from \( 12^{-5} \)).
• The second exponent is \( 7 \) (from \( 12^7 \)).

Now let's add the exponents together:

\[ -5 + 7 = 2 \]

So using the product rule, we can rewrite our expression:

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

Now we just need to find what \( 12^2 \) is:

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

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

In summary, I used the product rule of exponents to change \( 12^{-5} \cdot 12^{7} \) into \( 12^{2} \) by adding the exponents, and then I calculated \( 12^2 \) which is \( 144 \).