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Popular Trigonometry >

100=211.49-20.96cosh(0.03291765x)

Solution

100 = 211.49 - 20.96 cosh (0.03291765 x)

Solution

x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096}), x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

Degrees

x = - 4099.95343 \dots^{\circ}, x = 4099.95343 \dots^{\circ}

Solution steps

100 = 211.49 - 20.96 cosh (0.03291765 x)

Switch sides

211.49 - 20.96 cosh (0.03291765 x) = 100

Rewrite using trig identities

211.49 - 20.96 cosh (0.03291765 x) = 100

Use the Hyperbolic identity:

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

211.49 - 20.96 \cdot \frac{e ^{0.03291765 x} + e ^{- 0.03291765 x}}{2} = 100

211.49 - 20.96 \cdot \frac{e ^{0.03291765 x} + e ^{- 0.03291765 x}}{2} = 100

211.49 - 20.96 \cdot \frac{e ^{0.03291765 x} + e ^{- 0.03291765 x}}{2} = 100 : x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096}), x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

211.49 - 20.96 \cdot \frac{e ^{0.03291765 x} + e ^{- 0.03291765 x}}{2} = 100

Apply exponent rules

211.49 - 20.96 \cdot \frac{e ^{0.03291765 x} + e ^{- 0.03291765 x}}{2} = 100

Apply exponent rule:

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

e^{0.03291765 x} = (e^{x})^{0.03291765}, e^{- 0.03291765 x} = (e^{x})^{- 0.03291765}

211.49 - 20.96 \cdot \frac{( e ^{x} ) ^{0.03291765} + ( e ^{x} ) ^{- 0.03291765}}{2} = 100

211.49 - 20.96 \cdot \frac{( e ^{x} ) ^{0.03291765} + ( e ^{x} ) ^{- 0.03291765}}{2} = 100

Rewrite the equation with

e^{x} = u

211.49 - 20.96 \cdot \frac{( u ) ^{0.03291765} + ( u ) ^{- 0.03291765}}{2} = 100

Solve

211.49 - 20.96 \cdot \frac{u ^{0.03291765} + u ^{- 0.03291765}}{2} = 100 : u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}, u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

211.49 - 20.96 \cdot \frac{u ^{0.03291765} + u ^{- 0.03291765}}{2} = 100

Expand

211.49 - 20.96 \cdot \frac{u ^{0.03291765} + u ^{- 0.03291765}}{2} : 211.49 - \frac{20.96 u ^{0.03291765}}{2} - \frac{20.96}{2 u ^{0.03291765}}

211.49 - 20.96 \cdot \frac{u ^{0.03291765} + u ^{- 0.03291765}}{2}

\frac{u ^{0.03291765} + u ^{- 0.03291765}}{2} = \frac{u ^{0.0658353} + 1}{2 u ^{0.03291765}}

\frac{u ^{0.03291765} + u ^{- 0.03291765}}{2}

Apply exponent rule:

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

= \frac{u ^{0.03291765} + \frac{1}{u ^{0.03291765}}}{2}

Join

u^{0.03291765} + \frac{1}{u ^{0.03291765}} : \frac{u ^{0.0658353} + 1}{u ^{0.03291765}}

u^{0.03291765} + \frac{1}{u ^{0.03291765}}

Convert element to fraction:

u^{0.03291765} = \frac{u ^{0.03291765} u ^{0.03291765}}{u ^{0.03291765}}

= \frac{u ^{0.03291765} u ^{0.03291765}}{u ^{0.03291765}} + \frac{1}{u ^{0.03291765}}

Since the denominators are equal, combine the fractions:

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

= \frac{u ^{0.03291765} u ^{0.03291765} + 1}{u ^{0.03291765}}

u^{0.03291765} u^{0.03291765} + 1 = u^{0.0658353} + 1

u^{0.03291765} u^{0.03291765} + 1

u^{0.03291765} u^{0.03291765} = u^{0.0658353}

u^{0.03291765} u^{0.03291765}

Apply exponent rule:

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

u^{0.03291765} u^{0.03291765} = u^{0.03291765 + 0.03291765}

= u^{0.03291765 + 0.03291765}

Add the numbers:

0.03291765 + 0.03291765 = 0.0658353

= u^{0.0658353}

= u^{0.0658353} + 1

= \frac{u ^{0.0658353} + 1}{u ^{0.03291765}}

= \frac{\frac{u ^{0.0658353} + 1}{u ^{0.03291765}}}{2}

Apply the fraction rule:

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

= \frac{u ^{0.0658353} + 1}{u ^{0.03291765} \cdot 2}

= 211.49 - 20.96 \cdot \frac{u ^{0.0658353} + 1}{2 u ^{0.03291765}}

20.96 \cdot \frac{u ^{0.0658353} + 1}{u ^{0.03291765} \cdot 2} = \frac{20.96 u ^{0.0658353} + 20.96}{2 u ^{0.03291765}}

20.96 \cdot \frac{u ^{0.0658353} + 1}{u ^{0.03291765} \cdot 2}

Multiply fractions:

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

= \frac{( u ^{0.0658353} + 1 ) \cdot 20.96}{u ^{0.03291765} \cdot 2}

Expand

(u^{0.0658353} + 1) \cdot 20.96 : 20.96 u^{0.0658353} + 20.96

(u^{0.0658353} + 1) \cdot 20.96

= 20.96 (u^{0.0658353} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 20.96, b = u^{0.0658353}, c = 1

= 20.96 u^{0.0658353} + 20.96 \cdot 1

= 20.96 u^{0.0658353} + 1 \cdot 20.96

Multiply the numbers:

1 \cdot 20.96 = 20.96

= 20.96 u^{0.0658353} + 20.96

= \frac{20.96 u ^{0.0658353} + 20.96}{2 u ^{0.03291765}}

= 211.49 - \frac{20.96 u ^{0.0658353} + 20.96}{2 u ^{0.03291765}}

Apply the fraction rule:

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

\frac{20.96 u ^{0.0658353} + 20.96}{u ^{0.03291765} \cdot 2} = - (\frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2}) - (\frac{20.96}{u ^{0.03291765} \cdot 2})

= 211.49 - (\frac{20.96 u ^{0.0658353}}{2 u ^{0.03291765}}) - (\frac{20.96}{2 u ^{0.03291765}})

Remove parentheses:

(a) = a

= 211.49 - \frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2} - \frac{20.96}{u ^{0.03291765} \cdot 2}

Cancel

\frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2} : \frac{20.96 u ^{0.03291765}}{2}

\frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2}

Cancel

\frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2} : \frac{20.96 u ^{0.03291765}}{2}

\frac{20.96 u ^{0.0658353}}{u ^{0.03291765} \cdot 2}

Apply exponent rule:

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

\frac{u ^{0.0658353}}{u ^{0.03291765}} = u^{0.0658353 - 0.03291765}

= \frac{20.96 u ^{0.0658353 - 0.03291765}}{2}

Subtract the numbers:

0.0658353 - 0.03291765 = 0.03291765

= \frac{20.96 u ^{0.03291765}}{2}

= \frac{20.96 u ^{0.03291765}}{2}

= 211.49 - \frac{20.96 u ^{0.03291765}}{2} - \frac{20.96}{2 u ^{0.03291765}}

211.49 - \frac{20.96 u ^{0.03291765}}{2} - \frac{20.96}{2 u ^{0.03291765}} = 100

Use the following exponent property

: a^{n} = (a^{\frac{1}{m}})^{(n \cdot m)}

u^{0.03291765} = (u^{\frac{1}{8901}})^{(0.03291765 \cdot 8901)}

211.49 - \frac{20.96 ( u ^{\frac{1}{8901}} ) ^{293}}{2} - \frac{20.96}{2 ( u ^{\frac{1}{8901}} ) ^{293}} = 100

Rewrite the equation with

u^{\frac{1}{8901}} = v

211.49 - \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = 100

Solve

211.49 - \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = 100

Move

211.49

to the right side

211.49 - \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = 100

Subtract

211.49

from both sides

211.49 - \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} - 211.49 = 100 - 211.49

Simplify

- \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = - 111.49

- \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = - 111.49

Multiply both sides by

v^{293}

- \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}} = - 111.49

Multiply both sides by

v^{293}

- \frac{20.96 v ^{293}}{2} v^{293} - \frac{20.96}{2 v ^{293}} v^{293} = - 111.49 v^{293}

Simplify

- \frac{20.96 v ^{293}}{2} v^{293} - \frac{20.96}{2 v ^{293}} v^{293} = - 111.49 v^{293}

Simplify

- \frac{20.96 v ^{293}}{2} v^{293} : - \frac{20.96 v ^{586}}{2}

- \frac{20.96 v ^{293}}{2} v^{293}

Multiply fractions:

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

= - \frac{20.96 v ^{293} v ^{293}}{2}

20.96 v^{293} v^{293} = 20.96 v^{586}

20.96 v^{293} v^{293}

Apply exponent rule:

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

v^{293} v^{293} = v^{293 + 293}

= 20.96 v^{293 + 293}

Add the numbers:

293 + 293 = 586

= 20.96 v^{586}

= - \frac{20.96 v ^{586}}{2}

Simplify

- \frac{20.96}{2 v ^{293}} v^{293} : - \frac{20.96}{2}

- \frac{20.96}{2 v ^{293}} v^{293}

Multiply fractions:

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

= - \frac{20.96 v ^{293}}{2 v ^{293}}

Cancel the common factor:

v^{293}

= - \frac{20.96}{2}

- \frac{20.96 v ^{586}}{2} - \frac{20.96}{2} = - 111.49 v^{293}

- \frac{20.96 v ^{586}}{2} - \frac{20.96}{2} = - 111.49 v^{293}

- \frac{20.96 v ^{586}}{2} - \frac{20.96}{2} = - 111.49 v^{293}

Solve

- \frac{20.96 v ^{586}}{2} - \frac{20.96}{2} = - 111.49 v^{293}

Multiply both sides by

100

- \frac{20.96 v ^{586}}{2} - \frac{20.96}{2} = - 111.49 v^{293}

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- \frac{20.96 v ^{586}}{2} \cdot 100 - \frac{20.96}{2} \cdot 100 = - 111.49 v^{293} \cdot 100

Refine

- 1048 v^{586} - 1048 = - 11149 v^{293}

- 1048 v^{586} - 1048 = - 11149 v^{293}

Move

11149 v^{293}

to the left side

- 1048 v^{586} - 1048 = - 11149 v^{293}

Add

11149 v^{293}

to both sides

- 1048 v^{586} - 1048 + 11149 v^{293} = - 11149 v^{293} + 11149 v^{293}

Simplify

- 1048 v^{586} - 1048 + 11149 v^{293} = 0

- 1048 v^{586} - 1048 + 11149 v^{293} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 1048 v^{586} + 11149 v^{293} - 1048 = 0

Rewrite the equation with

u = v^{293}

and

u^{2} = v^{586}

- 1048 u^{2} + 11149 u - 1048 = 0

Solve

- 1048 u^{2} + 11149 u - 1048 = 0 : u = \frac{11149 - 119906985}{2096}, u = \frac{11149 + 119906985}{2096}

- 1048 u^{2} + 11149 u - 1048 = 0

Solve with the quadratic formula

- 1048 u^{2} + 11149 u - 1048 = 0

Quadratic Equation Formula:

For

a = - 1048, b = 11149, c = - 1048

u_{1, 2} = \frac{- 11149 \pm 1114 9 ^{2} - 4 ( - 1048 ) ( - 1048 )}{2 ( - 1048 )}

u_{1, 2} = \frac{- 11149 \pm 1114 9 ^{2} - 4 ( - 1048 ) ( - 1048 )}{2 ( - 1048 )}

1114 9^{2} - 4 (- 1048) (- 1048) = 119906985

1114 9^{2} - 4 (- 1048) (- 1048)

Apply rule

- (- a) = a

= 1114 9^{2} - 4 \cdot 1048 \cdot 1048

Multiply the numbers:

4 \cdot 1048 \cdot 1048 = 4393216

= 1114 9^{2} - 4393216

1114 9^{2} = 124300201

= 124300201 - 4393216

Subtract the numbers:

124300201 - 4393216 = 119906985

= 119906985

u_{1, 2} = \frac{- 11149 \pm 119906985}{2 ( - 1048 )}

Separate the solutions

u_{1} = \frac{- 11149 + 119906985}{2 ( - 1048 )}, u_{2} = \frac{- 11149 - 119906985}{2 ( - 1048 )}

u = \frac{- 11149 + 119906985}{2 ( - 1048 )} : \frac{11149 - 119906985}{2096}

\frac{- 11149 + 119906985}{2 ( - 1048 )}

Remove parentheses:

(- a) = - a

= \frac{- 11149 + 119906985}{- 2 \cdot 1048}

Multiply the numbers:

2 \cdot 1048 = 2096

= \frac{- 11149 + 119906985}{- 2096}

Apply the fraction rule:

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

- 11149 + 119906985 = - (11149 - 119906985)

= \frac{11149 - 119906985}{2096}

u = \frac{- 11149 - 119906985}{2 ( - 1048 )} : \frac{11149 + 119906985}{2096}

\frac{- 11149 - 119906985}{2 ( - 1048 )}

Remove parentheses:

(- a) = - a

= \frac{- 11149 - 119906985}{- 2 \cdot 1048}

Multiply the numbers:

2 \cdot 1048 = 2096

= \frac{- 11149 - 119906985}{- 2096}

Apply the fraction rule:

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

- 11149 - 119906985 = - (11149 + 119906985)

= \frac{11149 + 119906985}{2096}

The solutions to the quadratic equation are:

u = \frac{11149 - 119906985}{2096}, u = \frac{11149 + 119906985}{2096}

u = \frac{11149 - 119906985}{2096}, u = \frac{11149 + 119906985}{2096}

Substitute back

u = v^{293},

solve for

v

Solve

v^{293} = \frac{11149 - 119906985}{2096}

For

x^{n} = f (a)

, n is odd, the solution is

Solve

v^{293} = \frac{11149 + 119906985}{2096}

For

x^{n} = f (a)

, n is odd, the solution is

The solutions are

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

211.49 - \frac{20.96 v ^{293}}{2} - \frac{20.96}{2 v ^{293}}

and compare to zero

Solve

2 v^{293} = 0 : v = 0

2 v^{293} = 0

Divide both sides by

2

2 v^{293} = 0

Divide both sides by

2

2 v^{293} = 0

Divide both sides by

2

\frac{2 v ^{293}}{2} = \frac{0}{2}

Simplify

v^{293} = 0

v^{293} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

Substitute back

v = u^{\frac{1}{8901}},

solve for

u

Solve

Take both sides of the equation to the power of

8901 : u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Expand

(u^{\frac{1}{8901}})^{8901} : u

(u^{\frac{1}{8901}})^{8901}

Apply exponent rule:

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

= u^{\frac{1}{8901} \cdot 8901}

\frac{1}{8901} \cdot 8901 = 1

\frac{1}{8901} \cdot 8901

Multiply fractions:

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

= \frac{1 \cdot 8901}{8901}

Cancel the common factor:

8901

= 1

= u

Expand

Apply radical rule:

= (\frac{11149 - 119906985}{2096})^{\frac{1}{293}}^{8901}

Apply exponent rule:

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

= (\frac{11149 - 119906985}{2096})^{\frac{1}{293} \cdot 8901}

\frac{1}{293} \cdot 8901 = \frac{8901}{293}

\frac{1}{293} \cdot 8901

Multiply fractions:

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

= \frac{1 \cdot 8901}{293}

Multiply the numbers:

1 \cdot 8901 = 8901

= \frac{8901}{293}

= (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Verify Solutions:

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}} :

True

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{\frac{1}{8901}}

Apply exponent rule:

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

= (\frac{11149 - 119906985}{2096})^{\frac{8901}{293} \cdot \frac{1}{8901}}

Refine

= (\frac{11149 - 119906985}{2096})^{\frac{1}{293}}

True

The solution is

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Solve

Take both sides of the equation to the power of

8901 : u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Expand

(u^{\frac{1}{8901}})^{8901} : u

(u^{\frac{1}{8901}})^{8901}

Apply exponent rule:

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

= u^{\frac{1}{8901} \cdot 8901}

\frac{1}{8901} \cdot 8901 = 1

\frac{1}{8901} \cdot 8901

Multiply fractions:

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

= \frac{1 \cdot 8901}{8901}

Cancel the common factor:

8901

= 1

= u

Expand

Apply radical rule:

= (\frac{11149 + 119906985}{2096})^{\frac{1}{293}}^{8901}

Apply exponent rule:

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

= (\frac{11149 + 119906985}{2096})^{\frac{1}{293} \cdot 8901}

\frac{1}{293} \cdot 8901 = \frac{8901}{293}

\frac{1}{293} \cdot 8901

Multiply fractions:

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

= \frac{1 \cdot 8901}{293}

Multiply the numbers:

1 \cdot 8901 = 8901

= \frac{8901}{293}

= (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Verify Solutions:

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

True

Check the solutions by plugging them into

Remove the ones that don't agree with the equation.

Plug in

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}} :

True

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{\frac{1}{8901}}

Apply exponent rule:

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

= (\frac{11149 + 119906985}{2096})^{\frac{8901}{293} \cdot \frac{1}{8901}}

Refine

= (\frac{11149 + 119906985}{2096})^{\frac{1}{293}}

True

The solution is

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}, u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Verify Solutions:

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

True

, u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

True

Check the solutions by plugging them into

211.49 - 20.96 \frac{u ^{0.03291765} + u ^{- 0.03291765}}{2} = 100

Remove the ones that don't agree with the equation.

Plug in

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}} :

True

211.49 - 20.96 \cdot \frac{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = 100

211.49 - 20.96 \cdot \frac{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = 99.99999 \dots

211.49 - 20.96 \cdot \frac{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2}

\frac{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = \frac{10.63835 \dots}{2}

\frac{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2}

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{0.03291765} = 0.09484 \dots

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{0.03291765}

Apply exponent rule:

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

= (\frac{11149 - 119906985}{2096})^{\frac{8901}{293} \cdot 0.03291765}

\frac{8901}{293} \cdot 0.03291765 = 1.00000 \dots

\frac{8901}{293} \cdot 0.03291765

Multiply fractions:

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

= \frac{8901 \cdot 0.03291765}{293}

Multiply the numbers:

8901 \cdot 0.03291765 = 293.00000 \dots

= \frac{293.00000 \dots}{293}

Divide the numbers:

\frac{293.00000 \dots}{293} = 1.00000 \dots

= 1.00000 \dots

= (\frac{11149 - 119906985}{2096})^{1.00000 \dots}

\frac{11149 - 119906985}{2096} = \frac{198.79520 \dots}{2096}

\frac{11149 - 119906985}{2096}

Convert element to a decimal form

119906985 = 10950.20479 \dots

= \frac{11149 - 10950.20479 \dots}{2096}

Subtract the numbers:

11149 - 10950.20479 \dots = 198.79520 \dots

= \frac{198.79520 \dots}{2096}

= (\frac{198.79520 \dots}{2096})^{1.00000 \dots}

Divide the numbers:

\frac{198.79520 \dots}{2096} = 0.09484 \dots

= 0.09484 \dots^{1.00000 \dots}

0.09484 \dots^{1.00000 \dots} = 0.09484 \dots

= 0.09484 \dots

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{- 0.03291765} = 10.54351 \dots

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{- 0.03291765}

Apply exponent rule:

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

= \frac{1}{( ( \frac{11149 - 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765}}

(\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}^{0.03291765} : (\frac{11149 - 119906985}{2096})^{1.00000 \dots}

Apply exponent rule:

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

= (\frac{11149 - 119906985}{2096})^{\frac{8901}{293} \cdot 0.03291765}

\frac{8901}{293} \cdot 0.03291765 = 1.00000 \dots

\frac{8901}{293} \cdot 0.03291765

Multiply fractions:

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

= \frac{8901 \cdot 0.03291765}{293}

Multiply the numbers:

8901 \cdot 0.03291765 = 293.00000 \dots

= \frac{293.00000 \dots}{293}

Divide the numbers:

\frac{293.00000 \dots}{293} = 1.00000 \dots

= 1.00000 \dots

= (\frac{11149 - 119906985}{2096})^{1.00000 \dots}

= \frac{1}{( \frac{11149 - 119906985}{2096} ) ^{1.00000 \dots}}

\frac{11149 - 119906985}{2096} = \frac{198.79520 \dots}{2096}

\frac{11149 - 119906985}{2096}

Convert element to a decimal form

119906985 = 10950.20479 \dots

= \frac{11149 - 10950.20479 \dots}{2096}

Subtract the numbers:

11149 - 10950.20479 \dots = 198.79520 \dots

= \frac{198.79520 \dots}{2096}

= \frac{1}{( \frac{198.79520 \dots}{2096} ) ^{1.00000 \dots}}

(\frac{198.79520 \dots}{2096})^{1.00000 \dots} = 0.09484 \dots

= \frac{1}{0.09484 \dots}

Divide the numbers:

\frac{1}{0.09484 \dots} = 10.54351 \dots

= 10.54351 \dots

= \frac{0.09484 \dots + 10.54351 \dots}{2}

Add the numbers:

0.09484 \dots + 10.54351 \dots = 10.63835 \dots

= \frac{10.63835 \dots}{2}

= 211.49 - 20.96 \cdot \frac{10.63835 \dots}{2}

20.96 \cdot \frac{10.63835 \dots}{2} = 111.49000 \dots

20.96 \cdot \frac{10.63835 \dots}{2}

Multiply fractions:

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

= \frac{10.63835 \dots \cdot 20.96}{2}

Multiply the numbers:

10.63835 \dots \cdot 20.96 = 222.98000 \dots

= \frac{222.98000 \dots}{2}

Divide the numbers:

\frac{222.98000 \dots}{2} = 111.49000 \dots

= 111.49000 \dots

= 211.49 - 111.49000 \dots

Subtract the numbers:

211.49 - 111.49000 \dots = 99.99999 \dots

= 99.99999 \dots

99.99999 \dots = 100

True

Plug in

u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}} :

True

211.49 - 20.96 \cdot \frac{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = 100

211.49 - 20.96 \cdot \frac{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = 99.99999 \dots

211.49 - 20.96 \cdot \frac{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2}

\frac{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2} = \frac{10.63835 \dots}{2}

\frac{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765} + ( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{- 0.03291765}}{2}

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{0.03291765} = 10.54351 \dots

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{0.03291765}

Apply exponent rule:

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

= (\frac{11149 + 119906985}{2096})^{\frac{8901}{293} \cdot 0.03291765}

\frac{8901}{293} \cdot 0.03291765 = 1.00000 \dots

\frac{8901}{293} \cdot 0.03291765

Multiply fractions:

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

= \frac{8901 \cdot 0.03291765}{293}

Multiply the numbers:

8901 \cdot 0.03291765 = 293.00000 \dots

= \frac{293.00000 \dots}{293}

Divide the numbers:

\frac{293.00000 \dots}{293} = 1.00000 \dots

= 1.00000 \dots

= (\frac{11149 + 119906985}{2096})^{1.00000 \dots}

\frac{11149 + 119906985}{2096} = \frac{22099.20479 \dots}{2096}

\frac{11149 + 119906985}{2096}

Convert element to a decimal form

119906985 = 10950.20479 \dots

= \frac{11149 + 10950.20479 \dots}{2096}

Add the numbers:

11149 + 10950.20479 \dots = 22099.20479 \dots

= \frac{22099.20479 \dots}{2096}

= (\frac{22099.20479 \dots}{2096})^{1.00000 \dots}

Divide the numbers:

\frac{22099.20479 \dots}{2096} = 10.54351 \dots

= 10.54351 \dots^{1.00000 \dots}

10.54351 \dots^{1.00000 \dots} = 10.54351 \dots

= 10.54351 \dots

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{- 0.03291765} = 0.09484 \dots

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{- 0.03291765}

Apply exponent rule:

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

= \frac{1}{( ( \frac{11149 + 119906985}{2096} ) ^{\frac{8901}{293}} ) ^{0.03291765}}

(\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}^{0.03291765} : (\frac{11149 + 119906985}{2096})^{1.00000 \dots}

Apply exponent rule:

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

= (\frac{11149 + 119906985}{2096})^{\frac{8901}{293} \cdot 0.03291765}

\frac{8901}{293} \cdot 0.03291765 = 1.00000 \dots

\frac{8901}{293} \cdot 0.03291765

Multiply fractions:

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

= \frac{8901 \cdot 0.03291765}{293}

Multiply the numbers:

8901 \cdot 0.03291765 = 293.00000 \dots

= \frac{293.00000 \dots}{293}

Divide the numbers:

\frac{293.00000 \dots}{293} = 1.00000 \dots

= 1.00000 \dots

= (\frac{11149 + 119906985}{2096})^{1.00000 \dots}

= \frac{1}{( \frac{11149 + 119906985}{2096} ) ^{1.00000 \dots}}

\frac{11149 + 119906985}{2096} = \frac{22099.20479 \dots}{2096}

\frac{11149 + 119906985}{2096}

Convert element to a decimal form

119906985 = 10950.20479 \dots

= \frac{11149 + 10950.20479 \dots}{2096}

Add the numbers:

11149 + 10950.20479 \dots = 22099.20479 \dots

= \frac{22099.20479 \dots}{2096}

= \frac{1}{( \frac{22099.20479 \dots}{2096} ) ^{1.00000 \dots}}

(\frac{22099.20479 \dots}{2096})^{1.00000 \dots} = 10.54351 \dots

= \frac{1}{10.54351 \dots}

Divide the numbers:

\frac{1}{10.54351 \dots} = 0.09484 \dots

= 0.09484 \dots

= \frac{10.54351 \dots + 0.09484 \dots}{2}

Add the numbers:

10.54351 \dots + 0.09484 \dots = 10.63835 \dots

= \frac{10.63835 \dots}{2}

= 211.49 - 20.96 \cdot \frac{10.63835 \dots}{2}

20.96 \cdot \frac{10.63835 \dots}{2} = 111.49000 \dots

20.96 \cdot \frac{10.63835 \dots}{2}

Multiply fractions:

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

= \frac{10.63835 \dots \cdot 20.96}{2}

Multiply the numbers:

10.63835 \dots \cdot 20.96 = 222.98000 \dots

= \frac{222.98000 \dots}{2}

Divide the numbers:

\frac{222.98000 \dots}{2} = 111.49000 \dots

= 111.49000 \dots

= 211.49 - 111.49000 \dots

Subtract the numbers:

211.49 - 111.49000 \dots = 99.99999 \dots

= 99.99999 \dots

99.99999 \dots = 100

True

The solutions are

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}, u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

u = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}, u = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}} : x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096})

e^{x} = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Apply exponent rules

e^{x} = (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}}

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (\frac{11149 - 119906985}{2096})^{\frac{8901}{293}} = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096})

Solve

e^{x} = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}} : x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

e^{x} = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Apply exponent rules

e^{x} = (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}}

Apply log rule:

ln (x^{a}) = a \cdot ln (x)

ln (\frac{11149 + 119906985}{2096})^{\frac{8901}{293}} = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096}), x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

x = \frac{8901}{293} ln (\frac{11149 - 119906985}{2096}), x = \frac{8901}{293} ln (\frac{11149 + 119906985}{2096})

Graph

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