What is the value of sinh(2*2.6953125-4)-0.7*2.6953125 ?

The value of sinh(2*2.6953125-4)-0.7*2.6953125 is (128(e^{89/32}-1)-483e^{89/64})/(256e^{89/64)}

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sinh(22.6953125-4)-0.72.6953125

Solution

sinh (2 \cdot 2.6953125 - 4) - 0.7 \cdot 2.6953125

Solution

\frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{256 e ^{\frac{89}{64}}}

Decimal Notation

- 0.00249 \dots

Solution steps

sinh (2 \cdot 2.6953125 - 4) - 0.7 \cdot 2.6953125

= sinh (2 \cdot \frac{345}{128} - 4) - \frac{7}{10} \cdot \frac{345}{128}

Simplify:

2 \cdot \frac{345}{128} - 4 = \frac{89}{64}

2 \cdot \frac{345}{128} - 4

2 \cdot \frac{345}{128} = \frac{345}{64}

2 \cdot \frac{345}{128}

Multiply fractions:

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

= \frac{345 \cdot 2}{128}

Multiply the numbers:

345 \cdot 2 = 690

= \frac{690}{128}

Cancel the common factor:

2

= \frac{345}{64}

= \frac{345}{64} - 4

Convert element to fraction:

4 = \frac{4 \cdot 64}{64}

= \frac{345}{64} - \frac{4 \cdot 64}{64}

Since the denominators are equal, combine the fractions:

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

= \frac{345 - 4 \cdot 64}{64}

345 - 4 \cdot 64 = 89

345 - 4 \cdot 64

Multiply the numbers:

4 \cdot 64 = 256

= 345 - 256

Subtract the numbers:

345 - 256 = 89

= 89

= \frac{89}{64}

sinh (\frac{89}{64}) - \frac{7}{10} \cdot \frac{345}{128} = \frac{256 sinh ( \frac{89}{64} ) - 483}{256}

sinh (\frac{89}{64}) - \frac{7}{10} \cdot \frac{345}{128}

\frac{7}{10} \cdot \frac{345}{128} = \frac{483}{256}

\frac{7}{10} \cdot \frac{345}{128}

Cross-cancel common factor:

5

345, 10

Greatest Common Divisor (GCD)

Prime factorization of

345 : 3 \cdot 5 \cdot 23

345

345

divides by

3345 = 115 \cdot 3

= 3 \cdot 115

115

divides by

5115 = 23 \cdot 5

= 3 \cdot 5 \cdot 23

3, 5, 23

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5 \cdot 23

Prime factorization of

10 : 2 \cdot 5

10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 5

The prime factors common to

345, 10

are

= 5

= \frac{7}{2} \cdot \frac{69}{128}

Multiply fractions:

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

= \frac{7 \cdot 69}{2 \cdot 128}

Multiply the numbers:

7 \cdot 69 = 483

= \frac{483}{2 \cdot 128}

Multiply the numbers:

2 \cdot 128 = 256

= \frac{483}{256}

= sinh (\frac{89}{64}) - \frac{483}{256}

Convert element to fraction:

sinh (\frac{89}{64}) = \frac{sinh ( \frac{89}{64} ) 256}{256}

= \frac{sinh ( \frac{89}{64} ) \cdot 256}{256} - \frac{483}{256}

Since the denominators are equal, combine the fractions:

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

= \frac{sinh ( \frac{89}{64} ) \cdot 256 - 483}{256}

= \frac{256 sinh ( \frac{89}{64} ) - 483}{256}

Rewrite using trig identities:

sinh (\frac{89}{64}) = \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}}

sinh (\frac{89}{64})

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

= \frac{e ^{\frac{89}{64}} - e ^{- \frac{89}{64}}}{2}

\frac{e ^{\frac{89}{64}} - e ^{- \frac{89}{64}}}{2} = \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}}

\frac{e ^{\frac{89}{64}} - e ^{- \frac{89}{64}}}{2}

Apply exponent rule:

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

= \frac{e ^{\frac{89}{64}} - \frac{1}{e ^{\frac{89}{64}}}}{2}

Join

e^{\frac{89}{64}} - \frac{1}{e ^{\frac{89}{64}}} : \frac{e ^{\frac{89}{32}} - 1}{e ^{\frac{89}{64}}}

e^{\frac{89}{64}} - \frac{1}{e ^{\frac{89}{64}}}

Convert element to fraction:

e^{\frac{89}{64}} = \frac{e ^{\frac{89}{64}} e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}

= \frac{e ^{\frac{89}{64}} e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}} - \frac{1}{e ^{\frac{89}{64}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{89}{64}} e ^{\frac{89}{64}} - 1}{e ^{\frac{89}{64}}}

e^{\frac{89}{64}} e^{\frac{89}{64}} - 1 = e^{2 \cdot \frac{89}{64}} - 1

e^{\frac{89}{64}} e^{\frac{89}{64}} - 1

e^{\frac{89}{64}} e^{\frac{89}{64}} = e^{2 \cdot \frac{89}{64}}

e^{\frac{89}{64}} e^{\frac{89}{64}}

Apply exponent rule:

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

e^{\frac{89}{64}} e^{\frac{89}{64}} = e^{\frac{89}{64} + \frac{89}{64}}

= e^{\frac{89}{64} + \frac{89}{64}}

Add similar elements:

\frac{89}{64} + \frac{89}{64} = 2 \cdot \frac{89}{64}

= e^{2 \cdot \frac{89}{64}}

= e^{2 \cdot \frac{89}{64}} - 1

= \frac{e ^{2 \cdot \frac{89}{64}} - 1}{e ^{\frac{89}{64}}}

Multiply

2 \cdot \frac{89}{64} : \frac{89}{32}

2 \cdot \frac{89}{64}

Multiply fractions:

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

= \frac{89 \cdot 2}{64}

Multiply the numbers:

89 \cdot 2 = 178

= \frac{178}{64}

Cancel the common factor:

2

= \frac{89}{32}

= \frac{e ^{\frac{89}{32}} - 1}{e ^{\frac{89}{64}}}

= \frac{\frac{e ^{\frac{89}{32}} - 1}{e ^{\frac{89}{64}}}}{2}

Apply the fraction rule:

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

= \frac{e ^{\frac{89}{32}} - 1}{e ^{\frac{89}{64}} \cdot 2}

= \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}}

= \frac{256 \cdot \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}} - 483}{256}

\frac{256 \cdot \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}} - 483}{256} = \frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{256 e ^{\frac{89}{64}}}

\frac{256 \cdot \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}} - 483}{256}

256 \cdot \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}} = \frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}}

256 \cdot \frac{e ^{\frac{89}{32}} - 1}{2 e ^{\frac{89}{64}}}

Multiply fractions:

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

= \frac{( e ^{\frac{89}{32}} - 1 ) \cdot 256}{2 e ^{\frac{89}{64}}}

Divide the numbers:

\frac{256}{2} = 128

= \frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}}

= \frac{\frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}} - 483}{256}

Join

\frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}} - 483 : \frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}

\frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}} - 483

Convert element to fraction:

483 = \frac{483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}

= \frac{128 ( e ^{\frac{89}{32}} - 1 )}{e ^{\frac{89}{64}}} - \frac{483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}

Since the denominators are equal, combine the fractions:

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

= \frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}

= \frac{\frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}}}}{256}

Apply the fraction rule:

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

= \frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{e ^{\frac{89}{64}} \cdot 256}

= \frac{128 ( e ^{\frac{89}{32}} - 1 ) - 483 e ^{\frac{89}{64}}}{256 e ^{\frac{89}{64}}}

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Frequently Asked Questions (FAQ)

What is the value of sinh(2*2.6953125-4)-0.7*2.6953125 ?
The value of sinh(2*2.6953125-4)-0.7*2.6953125 is (128(e^{89/32}-1)-483e^{89/64})/(256e^{89/64)}