What is the value of tanh(5) ?

The value of tanh(5) is (e^{10}-1)/(e^{10)+1}

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tanh(5)

tanh (5)

\frac{e ^{10} - 1}{e ^{10} + 1}

Decimal Notation

0.99990 \dots

Solution steps

tanh (5)

Use the Hyperbolic identity:

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

= \frac{e ^{5} - e ^{- 5}}{e ^{5} + e ^{- 5}}

\frac{e ^{5} - e ^{- 5}}{e ^{5} + e ^{- 5}} = \frac{e ^{10} - 1}{e ^{10} + 1}

\frac{e ^{5} - e ^{- 5}}{e ^{5} + e ^{- 5}}

Apply exponent rule:

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

e^{- 5} = \frac{1}{e ^{5}}

= \frac{e ^{5} - \frac{1}{e ^{5}}}{e ^{5} + \frac{1}{e ^{5}}}

Join

e^{5} + \frac{1}{e ^{5}} : \frac{e ^{10} + 1}{e ^{5}}

e^{5} + \frac{1}{e ^{5}}

Convert element to fraction:

e^{5} = \frac{e ^{5} e ^{5}}{e ^{5}}

= \frac{e ^{5} e ^{5}}{e ^{5}} + \frac{1}{e ^{5}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{5} e ^{5} + 1}{e ^{5}}

e^{5} e^{5} + 1 = e^{10} + 1

e^{5} e^{5} + 1

e^{5} e^{5} = e^{10}

e^{5} e^{5}

Apply exponent rule:

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

e^{5} e^{5} = e^{5 + 5}

= e^{5 + 5}

Add the numbers:

5 + 5 = 10

= e^{10}

= e^{10} + 1

= \frac{e ^{10} + 1}{e ^{5}}

= \frac{e ^{5} - \frac{1}{e ^{5}}}{\frac{e ^{10} + 1}{e ^{5}}}

Join

e^{5} - \frac{1}{e ^{5}} : \frac{e ^{10} - 1}{e ^{5}}

e^{5} - \frac{1}{e ^{5}}

Convert element to fraction:

e^{5} = \frac{e ^{5} e ^{5}}{e ^{5}}

= \frac{e ^{5} e ^{5}}{e ^{5}} - \frac{1}{e ^{5}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{5} e ^{5} - 1}{e ^{5}}

e^{5} e^{5} - 1 = e^{10} - 1

e^{5} e^{5} - 1

e^{5} e^{5} = e^{10}

e^{5} e^{5}

Apply exponent rule:

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

e^{5} e^{5} = e^{5 + 5}

= e^{5 + 5}

Add the numbers:

5 + 5 = 10

= e^{10}

= e^{10} - 1

= \frac{e ^{10} - 1}{e ^{5}}

= \frac{\frac{e ^{10} - 1}{e ^{5}}}{\frac{e ^{10} + 1}{e ^{5}}}

Divide fractions:

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

= \frac{( e ^{10} - 1 ) e ^{5}}{e ^{5} ( e ^{10} + 1 )}

Cancel the common factor:

e^{5}

= \frac{e ^{10} - 1}{e ^{10} + 1}

= \frac{e ^{10} - 1}{e ^{10} + 1}

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