What is the value of arcsinh((e^8-1)/(2e^4)) ?

The value of arcsinh((e^8-1)/(2e^4)) is ln((e^8-1+sqrt(e^{16)+2e^8+1})/(2e^4))

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arcsinh((e^8-1)/(2e^4))

Solution

arcsinh (\frac{e ^{8} - 1}{2 e ^{4}})

Solution

ln (\frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}})

Decimal Notation

4

Solution steps

arcsinh (\frac{e ^{8} - 1}{2 e ^{4}})

Use the Hyperbolic identity:

arcsinh (x) = ln (x + x^{2} + 1)

= ln \frac{e ^{8} - 1}{2 e ^{4}} + (\frac{e ^{8} - 1}{2 e ^{4}})^{2} + 1

ln \frac{e ^{8} - 1}{2 e ^{4}} + (\frac{e ^{8} - 1}{2 e ^{4}})^{2} + 1 = ln (\frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}})

ln \frac{e ^{8} - 1}{2 e ^{4}} + (\frac{e ^{8} - 1}{2 e ^{4}})^{2} + 1

(\frac{e ^{8} - 1}{2 e ^{4}})^{2} + 1 = \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{4}}

(\frac{e ^{8} - 1}{2 e ^{4}})^{2} + 1

(\frac{e ^{8} - 1}{2 e ^{4}})^{2} = \frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}}

(\frac{e ^{8} - 1}{2 e ^{4}})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( e ^{8} - 1 ) ^{2}}{( 2 e ^{4} ) ^{2}}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

(2 e^{4})^{2} = 2^{2} (e^{4})^{2}

= \frac{( e ^{8} - 1 ) ^{2}}{2 ^{2} ( e ^{4} ) ^{2}}

(e^{4})^{2} : e^{8}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= e^{8}

= \frac{( e ^{8} - 1 ) ^{2}}{2 ^{2} e ^{8}}

2^{2} = 4

= \frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}}

= \frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}} + 1

Join

\frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}} + 1 : \frac{e ^{16} + 2 e ^{8} + 1}{4 e ^{8}}

\frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}} + 1

Convert element to fraction:

1 = \frac{1 \cdot 4 e ^{8}}{4 e ^{8}}

= \frac{( e ^{8} - 1 ) ^{2}}{4 e ^{8}} + \frac{1 \cdot 4 e ^{8}}{4 e ^{8}}

Since the denominators are equal, combine the fractions:

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

= \frac{( e ^{8} - 1 ) ^{2} + 1 \cdot 4 e ^{8}}{4 e ^{8}}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{( e ^{8} - 1 ) ^{2} + 4 e ^{8}}{4 e ^{8}}

Expand

(e^{8} - 1)^{2} + 4 e^{8} : e^{16} + 2 e^{8} + 1

(e^{8} - 1)^{2} + 4 e^{8}

(e^{8} - 1)^{2} : e^{16} - 2 e^{8} + 1

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = e^{8}, b = 1

= (e^{8})^{2} - 2 e^{8} \cdot 1 + 1^{2}

Simplify

(e^{8})^{2} - 2 e^{8} \cdot 1 + 1^{2} : e^{16} - 2 e^{8} + 1

(e^{8})^{2} - 2 e^{8} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (e^{8})^{2} - 2 \cdot 1 \cdot e^{8} + 1

(e^{8})^{2} = e^{16}

(e^{8})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{8 \cdot 2}

Multiply the numbers:

8 \cdot 2 = 16

= e^{16}

2 e^{8} \cdot 1 = 2 e^{8}

2 e^{8} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 e^{8}

= e^{16} - 2 e^{8} + 1

= e^{16} - 2 e^{8} + 1

= e^{16} - 2 e^{8} + 1 + 4 e^{8}

Simplify

e^{16} - 2 e^{8} + 1 + 4 e^{8} : e^{16} + 2 e^{8} + 1

e^{16} - 2 e^{8} + 1 + 4 e^{8}

Group like terms

= e^{16} - 2 e^{8} + 4 e^{8} + 1

Add similar elements:

- 2 e^{8} + 4 e^{8} = 2 e^{8}

= e^{16} + 2 e^{8} + 1

= e^{16} + 2 e^{8} + 1

= \frac{e ^{16} + 2 e ^{8} + 1}{4 e ^{8}}

= \frac{e ^{16} + 2 e ^{8} + 1}{4 e ^{8}}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{e ^{16} + 2 e ^{8} + 1}{4 e ^{8}}

Apply radical rule:

assuming

a \geq 0, b \geq 0

4 e^{8} = 4 e^{8}

= \frac{e ^{16} + 2 e ^{8} + 1}{4 e ^{8}}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{8}}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{8} = e^{4 \cdot 2} = (e^{4})^{2}

= \frac{e ^{16} + 2 e ^{8} + 1}{2 ( e ^{4} ) ^{2}}

Apply radical rule:

assuming

a \geq 0

(e^{4})^{2} = e^{4}

= \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{4}}

= ln (\frac{e ^{8} - 1}{2 e ^{4}} + \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{4}})

Join

\frac{e ^{8} - 1}{2 e ^{4}} + \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{4}} : \frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}}

\frac{e ^{8} - 1}{2 e ^{4}} + \frac{e ^{16} + 2 e ^{8} + 1}{2 e ^{4}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}}

= ln (\frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}})

= ln (\frac{e ^{8} - 1 + e ^{16} + 2 e ^{8} + 1}{2 e ^{4}})

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Frequently Asked Questions (FAQ)

What is the value of arcsinh((e^8-1)/(2e^4)) ?
The value of arcsinh((e^8-1)/(2e^4)) is ln((e^8-1+sqrt(e^{16)+2e^8+1})/(2e^4))