What is the value of tanh(e) ?

The value of tanh(e) is (e^{2e}-1)/(e^{2e)+1}

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tanh(e)

tanh (e)

\frac{e ^{2 e} - 1}{e ^{2 e} + 1}

Decimal Notation

0.99132 \dots

Solution steps

tanh (e)

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

= \frac{e ^{e} - e ^{- e}}{e ^{e} + e ^{- e}}

\frac{e ^{e} - e ^{- e}}{e ^{e} + e ^{- e}} = \frac{e ^{2 e} - 1}{e ^{2 e} + 1}

\frac{e ^{e} - e ^{- e}}{e ^{e} + e ^{- e}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

e^{- e} = \frac{1}{e ^{e}}

= \frac{e ^{e} - \frac{1}{e ^{e}}}{e ^{e} + \frac{1}{e ^{e}}}

Join

e^{e} + \frac{1}{e ^{e}} : \frac{e ^{2 e} + 1}{e ^{e}}

e^{e} + \frac{1}{e ^{e}}

Convert element to fraction:

e^{e} = \frac{e ^{e} e ^{e}}{e ^{e}}

= \frac{e ^{e} e ^{e}}{e ^{e}} + \frac{1}{e ^{e}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{e ^{e} e ^{e} + 1}{e ^{e}}

e^{e} e^{e} + 1 = e^{2 e} + 1

e^{e} e^{e} + 1

e^{e} e^{e} = e^{2 e}

e^{e} e^{e}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{e} e^{e} = e^{e + e}

= e^{e + e}

Add similar elements:

e + e = 2 e

= e^{2 e}

= e^{2 e} + 1

= \frac{e ^{2 e} + 1}{e ^{e}}

= \frac{e ^{e} - \frac{1}{e ^{e}}}{\frac{e ^{2 e} + 1}{e ^{e}}}

Join

e^{e} - \frac{1}{e ^{e}} : \frac{e ^{2 e} - 1}{e ^{e}}

e^{e} - \frac{1}{e ^{e}}

Convert element to fraction:

e^{e} = \frac{e ^{e} e ^{e}}{e ^{e}}

= \frac{e ^{e} e ^{e}}{e ^{e}} - \frac{1}{e ^{e}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{e ^{e} e ^{e} - 1}{e ^{e}}

e^{e} e^{e} - 1 = e^{2 e} - 1

e^{e} e^{e} - 1

e^{e} e^{e} = e^{2 e}

e^{e} e^{e}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{e} e^{e} = e^{e + e}

= e^{e + e}

Add similar elements:

e + e = 2 e

= e^{2 e}

= e^{2 e} - 1

= \frac{e ^{2 e} - 1}{e ^{e}}

= \frac{\frac{e ^{2 e} - 1}{e ^{e}}}{\frac{e ^{2 e} + 1}{e ^{e}}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{( e ^{2 e} - 1 ) e ^{e}}{e ^{e} ( e ^{2 e} + 1 )}

Cancel the common factor:

e^{e}

= \frac{e ^{2 e} - 1}{e ^{2 e} + 1}

= \frac{e ^{2 e} - 1}{e ^{2 e} + 1}

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