Apply the product rule of integer exponents to find the numerical equivalent of 12^−5⋅12^7. Show your work and explain how you used the product rule of integer exponents. (Make it like a 5 grader can understand)

Answer 1

To solve \( 12^{-5} \cdot 12^{7} \) using the product rule of integer exponents, let's first explain what the product rule is.

The product rule of integer exponents says that when you multiply two numbers that have the same base (in this case, the base is 12), you can add their exponents together. The rule can be written like this:

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

Here, \( a \) is the base, and \( m \) and \( n \) are the exponents.

Now let's apply this rule to our problem:

We have:

• The first term is \( 12^{-5} \). Here, the exponent is -5.
• The second term is \( 12^{7} \). Here, the exponent is 7.

Using the product rule, we will add the exponents:

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

Now we need to add the exponents \(-5\) and \(7\):

\[ -5 + 7 = 2 \]

So now we have:

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

Next, we need to find the numerical value of \( 12^{2} \). To do this, we multiply 12 by itself:

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

So the final answer is:

\[ 12^{-5} \cdot 12^{7} = 144 \]

Summary:

• We used the product rule of exponents to add the exponents \(-5\) and \(7\) together.
• We found that \( 12^{-5} \cdot 12^{7} = 12^{2} \).
• Finally, we calculated \( 12^{2} = 144 \).