To simplify the given expression, [tex]\(e^{a x} \cdot e^{-a x} - e^{2 a x}\)[/tex], let's follow the steps accordingly:
1. Combine the exponents in the first term:
The expression [tex]\(e^{a x} \cdot e^{-a x}\)[/tex] can be simplified using the properties of exponents. Specifically, the property [tex]\(e^{m} \cdot e^{n} = e^{m+n}\)[/tex]:
[tex]\[
e^{a x} \cdot e^{-a x} = e^{(a x) + (-a x)} = e^{0} = 1
\][/tex]
2. Substitute the simplified term back into the expression:
Now that we've simplified [tex]\(e^{a x} \cdot e^{-a x}\)[/tex] to 1, we substitute this result into the original expression:
[tex]\[
1 - e^{2 a x}
\][/tex]
3. Conclusion:
The simplified form of the given expression is:
[tex]\[
1 - e^{2 a x}
\][/tex]
Given the provided answer options:
- A [tex]\(e^{-2 a x}\)[/tex]
- B [tex]\(e - e^{2 a x}\)[/tex]
- C [tex]\(1 - e ^{2 a x}\)[/tex]
- D [tex]\(e^{-2 a 2 x} - e^{2 a x}\)[/tex]
The correct answer is:
[tex]\[ \boxed{C} \][/tex]
