Calculus Examples

${(e^{t} - e^{- t})}^{2}$

Step 1

Rewrite ${(e^{t} - e^{- t})}^{2}$ as $(e^{t} - e^{- t}) (e^{t} - e^{- t})$ .

$(e^{t} - e^{- t}) (e^{t} - e^{- t})$

Step 2

Expand $(e^{t} - e^{- t}) (e^{t} - e^{- t})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$e^{t} (e^{t} - e^{- t}) - e^{- t} (e^{t} - e^{- t})$

Step 2.2

Apply the distributive property.

$e^{t} e^{t} + e^{t} (- e^{- t}) - e^{- t} (e^{t} - e^{- t})$

Step 2.3

Apply the distributive property.

$e^{t} e^{t} + e^{t} (- e^{- t}) - e^{- t} e^{t} - e^{- t} (- e^{- t})$

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 $e^{t}$ by $e^{t}$ by adding the exponents.

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Step 3.1.1.1

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

$e^{t + t} + e^{t} (- e^{- t}) - e^{- t} e^{t} - e^{- t} (- e^{- t})$

Step 3.1.1.2

Add $t$ and $t$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

$e^{2 t} - e^{t} e^{- t} - e^{- t} e^{t} - e^{- t} (- e^{- t})$

Step 3.1.3

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

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Step 3.1.3.1

Move $e^{- t}$ .

$e^{2 t} - (e^{- t} e^{t}) - e^{- t} e^{t} - e^{- t} (- e^{- t})$

Step 3.1.3.2

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

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

Step 3.1.3.3

Add $- t$ and $t$ .

$e^{2 t} - e^{0} - e^{- t} e^{t} - e^{- t} (- e^{- t})$

Step 3.1.4

Simplify $- e^{0}$ .

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

Step 3.1.5

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

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Step 3.1.5.1

Move $e^{t}$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Subtract $t$ from $t$ .

$e^{2 t} - 1 - e^{0} - e^{- t} (- e^{- t})$

Step 3.1.6

Simplify $- e^{0}$ .

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

Step 3.1.7

Rewrite using the commutative property of multiplication.

$e^{2 t} - 1 - 1 - 1 \cdot - 1 e^{- t} e^{- t}$

Step 3.1.8

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

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Step 3.1.8.1

Move $e^{- t}$ .

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

Step 3.1.8.2

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

$e^{2 t} - 1 - 1 - 1 \cdot - 1 e^{- t - t}$

Step 3.1.8.3

Subtract $t$ from $- t$ .

$e^{2 t} - 1 - 1 - 1 \cdot - 1 e^{- 2 t}$

Step 3.1.9

Multiply $- 1$ by $- 1$ .

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

Step 3.1.10

Multiply $e^{- 2 t}$ by $1$ .

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

Step 3.2

Subtract $1$ from $- 1$ .

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