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Solved on Jan 03, 2024

Find the difference between $e^{-x}$ and $e^{-2x}$ .

STEP 1

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

STEP 2

Recognize that $e^{-x}$ and $e^{-2x}$ are exponential functions with the same base $e$ .

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

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