Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007?(1 point) Responses 200^15200 10,000^15 10,000 100^15 100^56

Answer 1

To use the Product Rule of Exponents, we apply the rule that states \( a^m \cdot a^n = a^{m+n} \).

First, let's recognize that \( 1008 \) and \( 1007 \) can be expressed in terms of base \( 10 \):

• \( 1008 = 10^2 \cdot 10^3 \) (which is \( 10^{2 + 3} = 10^5 \))
• \( 1007 = 10^2 \cdot 10^3 \) (which is also \( 10^{2 + 3} = 10^5 \))

So we can calculate:

\[ 1008 \cdot 1007 = 10^5 \cdot 10^5 = 10^{5 + 5} = 10^{10} \]

Now we can evaluate the provided choices:

1. \(200^{15}\)
2. \(10,000^{15}\) (which is \(10^{4 \cdot 15} = 10^{60}\))
3. \(10,000\) (which is \(10^4\))
4. \(100^{15}\) (which is \(10^{2 \cdot 15} = 10^{30}\))
5. \(100^{56}\) (which is \(10^{2 \cdot 56} = 10^{112}\))

None of the options directly simplify to the correct expression \(10^{10}\). Therefore, it appears there are no correct simplifications given among the responses.

If the intention was for options that simplify correctly according to the product rule, please double-check your options or clarify if there's a need for more context.