Solved on Jan 03, 2024

Find the difference between exe^{-x} and e2xe^{-2x}.

STEP 1

Assumptions
1. We are given the expression exe2xe^{-x} - e^{-2x}.
2. We are to simplify the expression if possible.

STEP 2

Recognize that exe^{-x} and e2xe^{-2x} are exponential functions with the same base ee.

STEP 3

Recall the property of exponents which states that eaeb=ea+be^{a} \cdot e^{b} = e^{a+b}.

STEP 4

Notice that e2xe^{-2x} can be written as (ex)2(e^{-x})^2 because exex=exx=e2xe^{-x} \cdot e^{-x} = e^{-x-x} = e^{-2x}.

STEP 5

Rewrite the expression using this property.
exe2x=ex(ex)2e^{-x} - e^{-2x} = e^{-x} - (e^{-x})^2

STEP 6

Factor out the common term exe^{-x} from the expression.
ex(ex)2=ex(1ex)e^{-x} - (e^{-x})^2 = e^{-x}(1 - e^{-x})

STEP 7

The expression is now simplified to ex(1ex)e^{-x}(1 - e^{-x}).
The simplified expression is ex(1ex)e^{-x}(1 - e^{-x}).

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