Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$e^{x} e^{x} + e^{x} e^{- x} + e^{- x} e^{x} + e^{- x} e^{- x}$

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 $e^{x}$ by $e^{x}$ 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.

$e^{x + x} + e^{x} e^{- x} + e^{- x} e^{x} + e^{- x} e^{- x}$

Step 3.1.1.2

Add $x$ and $x$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Subtract $x$ from $x$ .

$e^{2 x} + e^{0} + e^{- x} e^{x} + e^{- x} e^{- x}$

Step 3.1.3

Simplify $e^{0}$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Add $- x$ and $x$ .

$e^{2 x} + 1 + e^{0} + e^{- x} e^{- x}$

Step 3.1.5

Simplify $e^{0}$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

Subtract $x$ from $- x$ .

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

Step 3.2

Add $1$ and $1$ .

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