Basic Math Examples

$(0.5 e^{2 t} + e^{2 t} + 0.5) (e^{- 2 t} - 1)$

Step 1

Add $0.5 e^{2 t}$ and $e^{2 t}$ .

$(1.5 e^{2 t} + 0.5) (e^{- 2 t} - 1)$

Step 2

Expand $(1.5 e^{2 t} + 0.5) (e^{- 2 t} - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1.5 e^{2 t} (e^{- 2 t} - 1) + 0.5 (e^{- 2 t} - 1)$

Step 2.2

Apply the distributive property.

$1.5 e^{2 t} e^{- 2 t} + 1.5 e^{2 t} \cdot - 1 + 0.5 (e^{- 2 t} - 1)$

Step 2.3

Apply the distributive property.

$1.5 e^{2 t} e^{- 2 t} + 1.5 e^{2 t} \cdot - 1 + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $e^{2 t}$ by $e^{- 2 t}$ by adding the exponents.

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Step 3.1.1.1

Move $e^{- 2 t}$ .

$1.5 (e^{- 2 t} e^{2 t}) + 1.5 e^{2 t} \cdot - 1 + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1.5 e^{- 2 t + 2 t} + 1.5 e^{2 t} \cdot - 1 + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3.1.1.3

Add $- 2 t$ and $2 t$ .

$1.5 e^{0} + 1.5 e^{2 t} \cdot - 1 + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3.1.2

Simplify $1.5 e^{0}$ .

$1.5 + 1.5 e^{2 t} \cdot - 1 + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3.1.3

Multiply $- 1$ by $1.5$ .

$1.5 - 1.5 e^{2 t} + 0.5 e^{- 2 t} + 0.5 \cdot - 1$

Step 3.1.4

Multiply $0.5$ by $- 1$ .

$1.5 - 1.5 e^{2 t} + 0.5 e^{- 2 t} - 0.5$

Step 3.2

Subtract $0.5$ from $1.5$ .

$1 - 1.5 e^{2 t} + 0.5 e^{- 2 t}$