Question

Which of the following correctly used the Product Rule of Exponents to simplify the expression 1008⋅1007
100
8
⋅
100
7
?(1 point)
Responses

10056
100
56
100 superscript 56 baseline

10015
100
15
100 superscript 15 baseline

10,00015
10
,
000
15
10,000 superscript 15 baseline

20015

Answer 1

To simplify the expression \(1008 \cdot 1007\) using the Product Rule of Exponents, we can rewrite \(1008\) and \(1007\) in terms of powers of \(10\):

\[
1008 = 10^{8} \quad \text{and} \quad 1007 = 10^{7}
\]

When we multiply these two exponential terms, we can use the Product Rule of Exponents, which states that \(a^m \cdot a^n = a^{m+n}\). Therefore,

\[
1008 \cdot 1007 = 10^{8} \cdot 10^{7} = 10^{8+7} = 10^{15}
\]

Since \(100 = 10^2\), we also express the result as:

\[
10^{15} = (10^2)^{15/2} = 100^{15}
\]

Thus, the expression simplifies to \(100^{15}\).

Looking at the options provided:

1. \(100^{56}\)
2. \(100^{15}\) (This is correct)
3. \(10,000^{15}\) (This is equivalent to \(100^{30}\))
4. \(200^{15}\)

The correct response using the Product Rule of Exponents is:

**100^{15}**