What is the general solution for 0.05=(sinh(x/2))/(sinh(x)) ?

The general solution for 0.05=(sinh(x/2))/(sinh(x)) is x=ln(199+60sqrt(11)),x=ln(199-60sqrt(11))

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0.05=(sinh(x/2))/(sinh(x))

Solution

0.05 = \frac{sinh ( \frac{x}{2} )}{sinh ( x )}

Solution

x = ln (199 + 6011), x = ln (199 - 6011)

Degrees

x = 342.99807 \dots^{\circ}, x = - 342.99807 \dots^{\circ}

Solution steps

0.05 = \frac{sinh ( \frac{x}{2} )}{sinh ( x )}

Switch sides

\frac{sinh ( \frac{x}{2} )}{sinh ( x )} = 0.05

Rewrite using trig identities

\frac{sinh ( \frac{x}{2} )}{sinh ( x )} = 0.05

Use the Hyperbolic identity:

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

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} = 0.05

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} = 0.05

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} = 0.05 : x = ln (199 + 6011), x = ln (199 - 6011)

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} = 0.05

Multiply both sides by

\frac{e ^{x} - e ^{- x}}{2}

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} \cdot \frac{e ^{x} - e ^{- x}}{2} = 0.05 \cdot \frac{e ^{x} - e ^{- x}}{2}

Simplify

\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2} = \frac{0.05 ( e ^{x} - e ^{- x} )}{2}

Apply exponent rules

\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2} = \frac{0.05 ( e ^{x} - e ^{- x} )}{2}

Apply exponent rule:

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

e^{\frac{x}{2}} = e^{x}, e^{- \frac{x}{2}} = (e^{x})^{- 0.5}, e^{- x} = (e^{x})^{- 1}

\frac{e ^{x} - ( e ^{x} ) ^{- 0.5}}{2} = \frac{0.05 ( e ^{x} - ( e ^{x} ) ^{- 1} )}{2}

\frac{e ^{x} - ( e ^{x} ) ^{- 0.5}}{2} = \frac{0.05 ( e ^{x} - ( e ^{x} ) ^{- 1} )}{2}

Rewrite the equation with

e^{x} = u

\frac{u - ( u ) ^{- 0.5}}{2} = \frac{0.05 ( u - ( u ) ^{- 1} )}{2}

Solve

\frac{u - u ^{- 0.5}}{2} = \frac{0.05 ( u - u ^{- 1} )}{2} : u = 1, u = 199 + 6011, u = 199 - 6011

\frac{u - u ^{- 0.5}}{2} = \frac{0.05 ( u - u ^{- 1} )}{2}

Multiply both sides by

2

\frac{u - u ^{- 0.5}}{2} \cdot 2 = \frac{0.05 ( u - u ^{- 1} )}{2} \cdot 2

Simplify

u - u^{- 0.5} = 0.05 (u - \frac{1}{u})

Expand

u - u^{- 0.5} : u - \frac{1}{u}

u - u^{- 0.5}

Apply exponent rule:

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

= u - \frac{1}{u}

Expand

0.05 (u - \frac{1}{u}) : 0.05 u - \frac{0.05}{u}

0.05 (u - \frac{1}{u})

Apply the distributive law:

a (b - c) = ab - a c

a = 0.05, b = u, c = \frac{1}{u}

= 0.05 u - 0.05 \cdot \frac{1}{u}

0.05 \cdot \frac{1}{u} = \frac{0.05}{u}

0.05 \cdot \frac{1}{u}

Multiply fractions:

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

= \frac{1 \cdot 0.05}{u}

Multiply the numbers:

1 \cdot 0.05 = 0.05

= \frac{0.05}{u}

= 0.05 u - \frac{0.05}{u}

u - \frac{1}{u} = 0.05 u - \frac{0.05}{u}

Rewrite the equation with

u = v

v - \frac{1}{v} = 0.05 v^{2} - \frac{0.05}{v ^{2}}

Solve

v - \frac{1}{v} = 0.05 v^{2} - \frac{0.05}{v ^{2}} : v = - 1, v = 1, v = 10 + 311, v = 10 - 311

v - \frac{1}{v} = 0.05 v^{2} - \frac{0.05}{v ^{2}}

Multiply by LCM

v - \frac{1}{v} = 0.05 v^{2} - \frac{0.05}{v ^{2}}

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

v \cdot 100 - \frac{1}{v} \cdot 100 = 0.05 v^{2} \cdot 100 - \frac{0.05}{v ^{2}} \cdot 100

Refine

100 v - \frac{100}{v} = 5 v^{2} - \frac{5}{v ^{2}}

Find Least Common Multiplier of

v, v^{2} : v^{2}

v, v^{2}

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

v

v^{2}

= v^{2}

Multiply by LCM=

v^{2}

100 v v^{2} - \frac{100}{v} v^{2} = 5 v^{2} v^{2} - \frac{5}{v ^{2}} v^{2}

Simplify

100 v v^{2} - \frac{100}{v} v^{2} = 5 v^{2} v^{2} - \frac{5}{v ^{2}} v^{2}

Simplify

100 v v^{2} : 100 v^{3}

100 v v^{2}

Apply exponent rule:

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

v v^{2} = v^{1 + 2}

= 100 v^{1 + 2}

Add the numbers:

1 + 2 = 3

= 100 v^{3}

Simplify

- \frac{100}{v} v^{2} : - 100 v

- \frac{100}{v} v^{2}

Multiply fractions:

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

= - \frac{100 v ^{2}}{v}

Cancel the common factor:

v

= - 100 v

Simplify

5 v^{2} v^{2} : 5 v^{4}

5 v^{2} v^{2}

Apply exponent rule:

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

v^{2} v^{2} = v^{2 + 2}

= 5 v^{2 + 2}

Add the numbers:

2 + 2 = 4

= 5 v^{4}

Simplify

- \frac{5}{v ^{2}} v^{2} : - 5

- \frac{5}{v ^{2}} v^{2}

Multiply fractions:

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

= - \frac{5 v ^{2}}{v ^{2}}

Cancel the common factor:

v^{2}

= - 5

100 v^{3} - 100 v = 5 v^{4} - 5

100 v^{3} - 100 v = 5 v^{4} - 5

100 v^{3} - 100 v = 5 v^{4} - 5

Solve

100 v^{3} - 100 v = 5 v^{4} - 5 : v = - 1, v = 1, v = 10 + 311, v = 10 - 311

100 v^{3} - 100 v = 5 v^{4} - 5

Switch sides

5 v^{4} - 5 = 100 v^{3} - 100 v

Move

100 v

to the left side

5 v^{4} - 5 = 100 v^{3} - 100 v

Add

100 v

to both sides

5 v^{4} - 5 + 100 v = 100 v^{3} - 100 v + 100 v

Simplify

5 v^{4} - 5 + 100 v = 100 v^{3}

5 v^{4} - 5 + 100 v = 100 v^{3}

Move

100 v^{3}

to the left side

5 v^{4} - 5 + 100 v = 100 v^{3}

Subtract

100 v^{3}

from both sides

5 v^{4} - 5 + 100 v - 100 v^{3} = 100 v^{3} - 100 v^{3}

Simplify

5 v^{4} - 5 + 100 v - 100 v^{3} = 0

5 v^{4} - 5 + 100 v - 100 v^{3} = 0

Write in the standard form

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

5 v^{4} - 100 v^{3} + 100 v - 5 = 0

Factor

5 v^{4} - 100 v^{3} + 100 v - 5 : 5 (v + 1) (v - 1) (v^{2} - 20 v + 1)

5 v^{4} - 100 v^{3} + 100 v - 5

Factor out common term

5 : 5 (v^{4} - 20 v^{3} + 20 v - 1)

5 v^{4} - 100 v^{3} + 100 v - 5

Rewrite

100

5 \cdot 20

Rewrite

100

5 \cdot 20

= 5 v^{4} - 5 \cdot 20 v^{3} + 5 \cdot 20 v - 5

Factor out common term

5

= 5 (v^{4} - 20 v^{3} + 20 v - 1)

= 5 (v^{4} - 20 v^{3} + 20 v - 1)

Factor

v^{4} - 20 v^{3} + 20 v - 1 : (v + 1) (v - 1) (v^{2} - 20 v + 1)

v^{4} - 20 v^{3} + 20 v - 1

Use the rational root theorem

a_{0} = 1, a_{n} = 1

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1

Therefore, check the following rational numbers:

\pm \frac{1}{1}

- \frac{1}{1}

is a root of the expression, so factor out

v + 1

= (v + 1) \frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1}

\frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1} = v^{3} - 21 v^{2} + 21 v - 1

\frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1}

Divide

\frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1} : \frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1} = v^{3} + \frac{- 21 v ^{3} + 20 v - 1}{v + 1}

Divide the leading coefficients of the numerator

v^{4} - 20 v^{3} + 20 v - 1

and the divisor

v + 1 : \frac{v ^{4}}{v} = v^{3}

Quotient = v^{3}

Multiply

v + 1

v^{3} : v^{4} + v^{3}

Subtract

v^{4} + v^{3}

from

v^{4} - 20 v^{3} + 20 v - 1

to get new remainder

Remainder = - 21 v^{3} + 20 v - 1

Therefore

\frac{v ^{4} - 20 v ^{3} + 20 v - 1}{v + 1} = v^{3} + \frac{- 21 v ^{3} + 20 v - 1}{v + 1}

= v^{3} + \frac{- 21 v ^{3} + 20 v - 1}{v + 1}

Divide

\frac{- 21 v ^{3} + 20 v - 1}{v + 1} : \frac{- 21 v ^{3} + 20 v - 1}{v + 1} = - 21 v^{2} + \frac{21 v ^{2} + 20 v - 1}{v + 1}

Divide the leading coefficients of the numerator

- 21 v^{3} + 20 v - 1

and the divisor

v + 1 : \frac{- 21 v ^{3}}{v} = - 21 v^{2}

Quotient = - 21 v^{2}

Multiply

v + 1

- 21 v^{2} : - 21 v^{3} - 21 v^{2}

Subtract

- 21 v^{3} - 21 v^{2}

from

- 21 v^{3} + 20 v - 1

to get new remainder

Remainder = 21 v^{2} + 20 v - 1

Therefore

\frac{- 21 v ^{3} + 20 v - 1}{v + 1} = - 21 v^{2} + \frac{21 v ^{2} + 20 v - 1}{v + 1}

= v^{3} - 21 v^{2} + \frac{21 v ^{2} + 20 v - 1}{v + 1}

Divide

\frac{21 v ^{2} + 20 v - 1}{v + 1} : \frac{21 v ^{2} + 20 v - 1}{v + 1} = 21 v + \frac{- v - 1}{v + 1}

Divide the leading coefficients of the numerator

21 v^{2} + 20 v - 1

and the divisor

v + 1 : \frac{21 v ^{2}}{v} = 21 v

Quotient = 21 v

Multiply

v + 1

21 v : 21 v^{2} + 21 v

Subtract

21 v^{2} + 21 v

from

21 v^{2} + 20 v - 1

to get new remainder

Remainder = - v - 1

Therefore

\frac{21 v ^{2} + 20 v - 1}{v + 1} = 21 v + \frac{- v - 1}{v + 1}

= v^{3} - 21 v^{2} + 21 v + \frac{- v - 1}{v + 1}

Divide

\frac{- v - 1}{v + 1} : \frac{- v - 1}{v + 1} = - 1

Divide the leading coefficients of the numerator

- v - 1

and the divisor

v + 1 : \frac{- v}{v} = - 1

Quotient = - 1

Multiply

v + 1

- 1 : - v - 1

Subtract

- v - 1

from

- v - 1

to get new remainder

Remainder = 0

Therefore

\frac{- v - 1}{v + 1} = - 1

= v^{3} - 21 v^{2} + 21 v - 1

= v^{3} - 21 v^{2} + 21 v - 1

Factor

v^{3} - 21 v^{2} + 21 v - 1 : (v - 1) (v^{2} - 20 v + 1)

v^{3} - 21 v^{2} + 21 v - 1

Use the rational root theorem

a_{0} = 1, a_{n} = 1

The dividers of

a_{0} : 1,

The dividers of

a_{n} : 1

Therefore, check the following rational numbers:

\pm \frac{1}{1}

\frac{1}{1}

is a root of the expression, so factor out

v - 1

= (v - 1) \frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1}

\frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1} = v^{2} - 20 v + 1

\frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1}

Divide

\frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1} : \frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1} = v^{2} + \frac{- 20 v ^{2} + 21 v - 1}{v - 1}

Divide the leading coefficients of the numerator

v^{3} - 21 v^{2} + 21 v - 1

and the divisor

v - 1 : \frac{v ^{3}}{v} = v^{2}

Quotient = v^{2}

Multiply

v - 1

v^{2} : v^{3} - v^{2}

Subtract

v^{3} - v^{2}

from

v^{3} - 21 v^{2} + 21 v - 1

to get new remainder

Remainder = - 20 v^{2} + 21 v - 1

Therefore

\frac{v ^{3} - 21 v ^{2} + 21 v - 1}{v - 1} = v^{2} + \frac{- 20 v ^{2} + 21 v - 1}{v - 1}

= v^{2} + \frac{- 20 v ^{2} + 21 v - 1}{v - 1}

Divide

\frac{- 20 v ^{2} + 21 v - 1}{v - 1} : \frac{- 20 v ^{2} + 21 v - 1}{v - 1} = - 20 v + \frac{v - 1}{v - 1}

Divide the leading coefficients of the numerator

- 20 v^{2} + 21 v - 1

and the divisor

v - 1 : \frac{- 20 v ^{2}}{v} = - 20 v

Quotient = - 20 v

Multiply

v - 1

- 20 v : - 20 v^{2} + 20 v

Subtract

- 20 v^{2} + 20 v

from

- 20 v^{2} + 21 v - 1

to get new remainder

Remainder = v - 1

Therefore

\frac{- 20 v ^{2} + 21 v - 1}{v - 1} = - 20 v + \frac{v - 1}{v - 1}

= v^{2} - 20 v + \frac{v - 1}{v - 1}

Divide

\frac{v - 1}{v - 1} : \frac{v - 1}{v - 1} = 1

Divide the leading coefficients of the numerator

v - 1

and the divisor

v - 1 : \frac{v}{v} = 1

Quotient = 1

Multiply

v - 1

1 : v - 1

Subtract

v - 1

from

v - 1

to get new remainder

Remainder = 0

Therefore

\frac{v - 1}{v - 1} = 1

= v^{2} - 20 v + 1

= v^{2} - 20 v + 1

= (v - 1) (v^{2} - 20 v + 1)

= (v + 1) (v - 1) (v^{2} - 20 v + 1)

= 5 (v + 1) (v - 1) (v^{2} - 20 v + 1)

5 (v + 1) (v - 1) (v^{2} - 20 v + 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

v + 1 = 0 or v - 1 = 0 or v^{2} - 20 v + 1 = 0

Solve

v + 1 = 0 : v = - 1

v + 1 = 0

Move

1

to the right side

v + 1 = 0

Subtract

1

from both sides

v + 1 - 1 = 0 - 1

Simplify

v = - 1

v = - 1

Solve

v - 1 = 0 : v = 1

v - 1 = 0

Move

1

to the right side

v - 1 = 0

Add

1

to both sides

v - 1 + 1 = 0 + 1

Simplify

v = 1

v = 1

Solve

v^{2} - 20 v + 1 = 0 : v = 10 + 311, v = 10 - 311

v^{2} - 20 v + 1 = 0

Solve with the quadratic formula

v^{2} - 20 v + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = - 20, c = 1

v_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 20)^{2} - 4 \cdot 1 \cdot 1 = 611

(- 20)^{2} - 4 \cdot 1 \cdot 1

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 20)^{2} = 2 0^{2}

= 2 0^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2 0^{2} - 4

2 0^{2} = 400

= 400 - 4

Subtract the numbers:

400 - 4 = 396

= 396

Prime factorization of

396 : 2^{2} \cdot 3^{2} \cdot 11

396

396

divides by

2396 = 198 \cdot 2

= 2 \cdot 198

198

divides by

2198 = 99 \cdot 2

= 2 \cdot 2 \cdot 99

99

divides by

399 = 33 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 11

= 2^{2} \cdot 3^{2} \cdot 11

= 2^{2} \cdot 3^{2} \cdot 11

Apply radical rule:

= 11 2^{2} 3^{2}

Apply radical rule:

2^{2} = 2

= 211 3^{2}

Apply radical rule:

3^{2} = 3

= 2 \cdot 311

Refine

= 611

v_{1, 2} = \frac{- ( - 20 ) \pm 6 11}{2 \cdot 1}

Separate the solutions

v_{1} = \frac{- ( - 20 ) + 6 11}{2 \cdot 1}, v_{2} = \frac{- ( - 20 ) - 6 11}{2 \cdot 1}

v = \frac{- ( - 20 ) + 6 11}{2 \cdot 1} : 10 + 311

\frac{- ( - 20 ) + 6 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{20 + 6 11}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{20 + 6 11}{2}

Factor

20 + 611 : 2 (10 + 311)

20 + 611

Rewrite as

= 2 \cdot 10 + 2 \cdot 311

Factor out common term

2

= 2 (10 + 311)

= \frac{2 ( 10 + 3 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 + 311

v = \frac{- ( - 20 ) - 6 11}{2 \cdot 1} : 10 - 311

\frac{- ( - 20 ) - 6 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{20 - 6 11}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{20 - 6 11}{2}

Factor

20 - 611 : 2 (10 - 311)

20 - 611

Rewrite as

= 2 \cdot 10 - 2 \cdot 311

Factor out common term

2

= 2 (10 - 311)

= \frac{2 ( 10 - 3 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 10 - 311

The solutions to the quadratic equation are:

v = 10 + 311, v = 10 - 311

The solutions are

v = - 1, v = 1, v = 10 + 311, v = 10 - 311

v = - 1, v = 1, v = 10 + 311, v = 10 - 311

Verify Solutions

Find undefined (singularity) points:

v = 0

Take the denominator(s) of

v - \frac{1}{v}

and compare to zero

v = 0

Take the denominator(s) of

0.05 v^{2} - \frac{0.05}{v ^{2}}

and compare to zero

Solve

v^{2} = 0 : v = 0

v^{2} = 0

Apply rule

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

v = 0

The following points are undefined

v = 0

Combine undefined points with solutions:

v = - 1, v = 1, v = 10 + 311, v = 10 - 311

v = - 1, v = 1, v = 10 + 311, v = 10 - 311

Substitute back

v = u,

solve for

u

Solve

u = - 1 :

No Solution for

u \in R

u = - 1

f (u)

cannot be negative for

u \in R

No Solution for u \in R

Solve

u = 1 : u = 1

u = 1

Square both sides:

u = 1

u = 1

(u)^{2} = 1^{2}

Expand

(u)^{2} : u

(u)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (u^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= u^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= u

Expand

1^{2} : 1

1^{2}

Apply rule

1^{a} = 1

= 1

u = 1

u = 1

Verify Solutions:

u = 1

True

Check the solutions by plugging them into

u = 1

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

Plug in

u = 1 :

True

1 = 1

1 = 1

1

Apply rule

1 = 1

= 1

1 = 1

True

The solution is

u = 1

Solve

u = 10 + 311 : u = 199 + 6011

u = 10 + 311

Square both sides:

u = 199 + 6011

u = 10 + 311

(u)^{2} = (10 + 311)^{2}

Expand

(u)^{2} : u

(u)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (u^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= u^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= u

Expand

(10 + 311)^{2} : 199 + 6011

(10 + 311)^{2}

Apply Perfect Square Formula:

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

a = 10, b = 311

= 1 0^{2} + 2 \cdot 10 \cdot 311 + (311)^{2}

Simplify

1 0^{2} + 2 \cdot 10 \cdot 311 + (311)^{2} : 199 + 6011

1 0^{2} + 2 \cdot 10 \cdot 311 + (311)^{2}

1 0^{2} = 100

1 0^{2}

1 0^{2} = 100

= 100

2 \cdot 10 \cdot 311 = 6011

2 \cdot 10 \cdot 311

Multiply the numbers:

2 \cdot 10 \cdot 3 = 60

= 6011

(311)^{2} = 99

(311)^{2}

Apply exponent rule:

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

= 3^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 3^{2} \cdot 11

3^{2} = 9

= 9 \cdot 11

Multiply the numbers:

9 \cdot 11 = 99

= 99

= 100 + 6011 + 99

Add the numbers:

100 + 99 = 199

= 199 + 6011

= 199 + 6011

u = 199 + 6011

u = 199 + 6011

Verify Solutions:

u = 199 + 6011

True

Check the solutions by plugging them into

u = 10 + 311

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

Plug in

u = 199 + 6011 :

True

199 + 6011 = 10 + 311

199 + 6011 = 311 + 10

199 + 6011

= 99 + 6011 + 100

= 9 \cdot 11 + 6011 + 100

= (9)^{2} (11)^{2} + 6011 + (100)^{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

100 = 10

100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

= 3^{2} (11)^{2} + 6011 + 1 0^{2}

2 \cdot 311 \cdot 10 = 6011

2 \cdot 311 \cdot 10

Multiply the numbers:

2 \cdot 3 \cdot 10 = 60

= 6011

= (311)^{2} + 2 \cdot 311 \cdot 10 + 1 0^{2}

Apply Perfect Square Formula:

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

(311)^{2} + 2 \cdot 311 \cdot 10 + 1 0^{2} = (311 + 10)^{2}

= (311 + 10)^{2}

Apply radical rule:

(311 + 10)^{2} = 311 + 10

= 311 + 10

311 + 10 = 10 + 311

True

The solution is

u = 199 + 6011

Solve

u = 10 - 311 : u = 199 - 6011

u = 10 - 311

Square both sides:

u = 199 - 6011

u = 10 - 311

(u)^{2} = (10 - 311)^{2}

Expand

(u)^{2} : u

(u)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (u^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= u^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= u

Expand

(10 - 311)^{2} : 199 - 6011

(10 - 311)^{2}

Apply Perfect Square Formula:

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

a = 10, b = 311

= 1 0^{2} - 2 \cdot 10 \cdot 311 + (311)^{2}

Simplify

1 0^{2} - 2 \cdot 10 \cdot 311 + (311)^{2} : 199 - 6011

1 0^{2} - 2 \cdot 10 \cdot 311 + (311)^{2}

1 0^{2} = 100

1 0^{2}

1 0^{2} = 100

= 100

2 \cdot 10 \cdot 311 = 6011

2 \cdot 10 \cdot 311

Multiply the numbers:

2 \cdot 10 \cdot 3 = 60

= 6011

(311)^{2} = 99

(311)^{2}

Apply exponent rule:

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

= 3^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 3^{2} \cdot 11

3^{2} = 9

= 9 \cdot 11

Multiply the numbers:

9 \cdot 11 = 99

= 99

= 100 - 6011 + 99

Add the numbers:

100 + 99 = 199

= 199 - 6011

= 199 - 6011

u = 199 - 6011

u = 199 - 6011

Verify Solutions:

u = 199 - 6011

True

Check the solutions by plugging them into

u = 10 - 311

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

Plug in

u = 199 - 6011 :

True

199 - 6011 = 10 - 311

199 - 6011 = 10 - 311

199 - 6011

= 99 - 6011 + 100

= 9 \cdot 11 - 6011 + 100

= (9)^{2} (11)^{2} - 6011 + (100)^{2}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

100 = 10

100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

= 3^{2} (11)^{2} - 6011 + 1 0^{2}

2 \cdot 311 \cdot 10 = 6011

2 \cdot 311 \cdot 10

Multiply the numbers:

2 \cdot 3 \cdot 10 = 60

= 6011

= (311)^{2} - 2 \cdot 311 \cdot 10 + 1 0^{2}

Apply Perfect Square Formula:

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

(311)^{2} - 2 \cdot 311 \cdot 10 + 1 0^{2} = (311 - 10)^{2}

= (311 - 10)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(311 - 10)^{2} = (10 - 311)^{2}

= (10 - 311)^{2}

Apply radical rule:

(10 - 311)^{2} = 10 - 311

= 10 - 311

10 - 311 = 10 - 311

True

The solution is

u = 199 - 6011

u = 1, u = 199 + 6011, u = 199 - 6011

Verify Solutions:

u = 1

True

, u = 199 + 6011

True

, u = 199 - 6011

True

Check the solutions by plugging them into

\frac{u - u ^{- 0.5}}{2} = \frac{0.05 ( u - u ^{- 1} )}{2}

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

Plug in

u = 1 :

True

\frac{1 - 1 ^{- 0.5}}{2} = \frac{0.05 ( 1 - 1 ^{- 1} )}{2}

\frac{1 - 1 ^{- 0.5}}{2} = 0

\frac{1 - 1 ^{- 0.5}}{2}

Apply rule

1^{a} = 1

1^{- 0.5} = 1

= \frac{1 - 1}{2}

Apply rule

1 = 1

= \frac{1 - 1}{2}

Subtract the numbers:

1 - 1 = 0

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

\frac{0.05 ( 1 - 1 ^{- 1} )}{2} = 0

\frac{0.05 ( 1 - 1 ^{- 1} )}{2}

Apply rule

1^{a} = 1

1^{- 1} = 1

= \frac{0.05 ( 1 - 1 )}{2}

0.05 (1 - 1) = 0

0.05 (1 - 1)

Subtract the numbers:

1 - 1 = 0

= 0 \cdot 0.05

Apply rule

0 \cdot a = 0

= 0

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

0 = 0

True

Plug in

u = 199 + 6011 :

True

\frac{199 + 60 11 - ( 199 + 60 11 ) ^{- 0.5}}{2} = \frac{0.05 ( ( 199 + 60 11 ) - ( 199 + 60 11 ) ^{- 1} )}{2}

\frac{199 + 60 11 - ( 199 + 60 11 ) ^{- 0.5}}{2} = 9.94987 \dots

\frac{199 + 60 11 - ( 199 + 60 11 ) ^{- 0.5}}{2}

199 + 6011 = 397.99748 \dots

199 + 6011

6011 = 198.99748 \dots

6011

Convert element to a decimal form

11 = 3.31662 \dots

= 60 \cdot 3.31662 \dots

Multiply the numbers:

60 \cdot 3.31662 \dots = 198.99748 \dots

= 198.99748 \dots

= 199 + 198.99748 \dots

Add the numbers:

199 + 198.99748 \dots = 397.99748 \dots

= 397.99748 \dots

(199 + 6011)^{- 0.5} = 397.99748 \dots^{- 0.5}

(199 + 6011)^{- 0.5}

6011 = 198.99748 \dots

6011

Convert element to a decimal form

11 = 3.31662 \dots

= 60 \cdot 3.31662 \dots

Multiply the numbers:

60 \cdot 3.31662 \dots = 198.99748 \dots

= 198.99748 \dots

= (199 + 198.99748 \dots)^{- 0.5}

Add the numbers:

199 + 198.99748 \dots = 397.99748 \dots

= 397.99748 \dots^{- 0.5}

= \frac{397.99748 \dots - 397.99748 \dots ^{- 0.5}}{2}

397.99748 \dots = 19.94987 \dots

= \frac{19.94987 \dots - 397.99748 \dots ^{- 0.5}}{2}

397.99748 \dots^{- 0.5} = 0.05012 \dots

= \frac{19.94987 \dots - 0.05012 \dots}{2}

Subtract the numbers:

19.94987 \dots - 0.05012 \dots = 19.89974 \dots

= \frac{19.89974 \dots}{2}

Divide the numbers:

\frac{19.89974 \dots}{2} = 9.94987 \dots

= 9.94987 \dots

\frac{0.05 ( ( 199 + 60 11 ) - ( 199 + 60 11 ) ^{- 1} )}{2} = 9.94987 \dots

\frac{0.05 ( ( 199 + 60 11 ) - ( 199 + 60 11 ) ^{- 1} )}{2}

Remove parentheses:

(a) = a

= \frac{0.05 ( 199 + 60 11 - ( 199 + 60 11 ) ^{- 1} )}{2}

Factor

0.05 (199 + 6011 - (199 + 6011)^{- 1}) : 611

0.05 (199 + 6011 - (199 + 6011)^{- 1})

Factor

199 + 6011 - (199 + 6011)^{- 1} : 12011

199 + 6011 - (199 + 6011)^{- 1}

Apply exponent rule:

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

(199 + 6011) = (199 + 6011)^{- 1} (199 + 6011)^{2}

= - (199 + 6011)^{- 1} + 199 + 6011

Factor out common term

(199 + 6011)^{- 1}

= (199 + 6011)^{- 1} (- 1 + (199 + 6011)^{2})

Factor

(199 + 6011)^{2} - 1 : 120 (660 + 19911)

- 1 + (199 + 6011)^{2}

(199 + 6011)^{2} : 79201 + 2388011

Apply Perfect Square Formula:

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

a = 199, b = 6011

= 19 9^{2} + 2 \cdot 199 \cdot 6011 + (6011)^{2}

Simplify

19 9^{2} + 2 \cdot 199 \cdot 6011 + (6011)^{2} : 79201 + 2388011

19 9^{2} + 2 \cdot 199 \cdot 6011 + (6011)^{2}

19 9^{2} = 39601

19 9^{2}

19 9^{2} = 39601

= 39601

2 \cdot 199 \cdot 6011 = 2388011

2 \cdot 199 \cdot 6011

Multiply the numbers:

2 \cdot 199 \cdot 60 = 23880

= 2388011

(6011)^{2} = 39600

(6011)^{2}

Apply exponent rule:

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

= 6 0^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 6 0^{2} \cdot 11

6 0^{2} = 3600

= 3600 \cdot 11

Multiply the numbers:

3600 \cdot 11 = 39600

= 39600

= 39601 + 2388011 + 39600

Add the numbers:

39601 + 39600 = 79201

= 79201 + 2388011

= 79201 + 2388011

= - 1 + 79201 + 2388011

Add/Subtract the numbers:

- 1 + 79201 = 79200

= 79200 + 2388011

Rewrite as

= 120 \cdot 660 + 120 \cdot 19911

Factor out common term

120

= 120 (660 + 19911)

= 120 (199 + 6011)^{- 1} (660 + 19911)

Refine

= 12011

= 0.05 \cdot 12011

Refine

= 611

= \frac{6 11}{2}

Divide the numbers:

\frac{6}{2} = 3

= 311

Convert element to a decimal form

11 = 3.31662 \dots

= 3 \cdot 3.31662 \dots

Multiply the numbers:

3 \cdot 3.31662 \dots = 9.94987 \dots

= 9.94987 \dots

9.94987 \dots = 9.94987 \dots

True

Plug in

u = 199 - 6011 :

True

\frac{199 - 60 11 - ( 199 - 60 11 ) ^{- 0.5}}{2} = \frac{0.05 ( ( 199 - 60 11 ) - ( 199 - 60 11 ) ^{- 1} )}{2}

\frac{199 - 60 11 - ( 199 - 60 11 ) ^{- 0.5}}{2} = - 9.94987 \dots

\frac{199 - 60 11 - ( 199 - 60 11 ) ^{- 0.5}}{2}

199 - 6011 = 0.00251 \dots

199 - 6011

6011 = 198.99748 \dots

6011

Convert element to a decimal form

11 = 3.31662 \dots

= 60 \cdot 3.31662 \dots

Multiply the numbers:

60 \cdot 3.31662 \dots = 198.99748 \dots

= 198.99748 \dots

= 199 - 198.99748 \dots

Subtract the numbers:

199 - 198.99748 \dots = 0.00251 \dots

= 0.00251 \dots

(199 - 6011)^{- 0.5} = 0.00251 \dots^{- 0.5}

(199 - 6011)^{- 0.5}

6011 = 198.99748 \dots

6011

Convert element to a decimal form

11 = 3.31662 \dots

= 60 \cdot 3.31662 \dots

Multiply the numbers:

60 \cdot 3.31662 \dots = 198.99748 \dots

= 198.99748 \dots

= (199 - 198.99748 \dots)^{- 0.5}

Subtract the numbers:

199 - 198.99748 \dots = 0.00251 \dots

= 0.00251 \dots^{- 0.5}

= \frac{0.00251 \dots - 0.00251 \dots ^{- 0.5}}{2}

0.00251 \dots = 0.05012 \dots

= \frac{0.05012 \dots - 0.00251 \dots ^{- 0.5}}{2}

0.00251 \dots^{- 0.5} = 19.94987 \dots

= \frac{0.05012 \dots - 19.94987 \dots}{2}

Subtract the numbers:

0.05012 \dots - 19.94987 \dots = - 19.89974 \dots

= \frac{- 19.89974 \dots}{2}

Apply the fraction rule:

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

= - \frac{19.89974 \dots}{2}

Divide the numbers:

\frac{19.89974 \dots}{2} = 9.94987 \dots

= - 9.94987 \dots

\frac{0.05 ( ( 199 - 60 11 ) - ( 199 - 60 11 ) ^{- 1} )}{2} = - 9.94987 \dots

\frac{0.05 ( ( 199 - 60 11 ) - ( 199 - 60 11 ) ^{- 1} )}{2}

Remove parentheses:

(a) = a

= \frac{0.05 ( 199 - 60 11 - ( 199 - 60 11 ) ^{- 1} )}{2}

Factor

0.05 (199 - 6011 - (199 - 6011)^{- 1}) : - 611

0.05 (199 - 6011 - (199 - 6011)^{- 1})

Factor

199 - 6011 - (199 - 6011)^{- 1} : - 12011

199 - 6011 - (199 - 6011)^{- 1}

Apply exponent rule:

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

(199 - 6011) = (199 - 6011)^{- 1} (199 - 6011)^{2}

= - (199 - 6011)^{- 1} + 199 - 6011

Factor out common term

(199 - 6011)^{- 1}

= (199 - 6011)^{- 1} (- 1 + (199 - 6011)^{2})

Factor

(199 - 6011)^{2} - 1 : 120 (660 - 19911)

- 1 + (199 - 6011)^{2}

(199 - 6011)^{2} : 79201 - 2388011

Apply Perfect Square Formula:

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

a = 199, b = 6011

= 19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2}

Simplify

19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2} : 79201 - 2388011

19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2}

19 9^{2} = 39601

19 9^{2}

19 9^{2} = 39601

= 39601

2 \cdot 199 \cdot 6011 = 2388011

2 \cdot 199 \cdot 6011

Multiply the numbers:

2 \cdot 199 \cdot 60 = 23880

= 2388011

(6011)^{2} = 39600

(6011)^{2}

Apply exponent rule:

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

= 6 0^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 6 0^{2} \cdot 11

6 0^{2} = 3600

= 3600 \cdot 11

Multiply the numbers:

3600 \cdot 11 = 39600

= 39600

= 39601 - 2388011 + 39600

Add the numbers:

39601 + 39600 = 79201

= 79201 - 2388011

= 79201 - 2388011

= - 1 + 79201 - 2388011

Add/Subtract the numbers:

- 1 + 79201 = 79200

= 79200 - 2388011

Rewrite as

= 120 \cdot 660 - 120 \cdot 19911

Factor out common term

120

= 120 (660 - 19911)

= 120 (199 - 6011)^{- 1} (660 - 19911)

Refine

= - 12011

= - 0.05 \cdot 12011

Refine

= - 611

= - \frac{6 11}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 311

Convert element to a decimal form

11 = 3.31662 \dots

= - 3 \cdot 3.31662 \dots

Multiply the numbers:

3 \cdot 3.31662 \dots = 9.94987 \dots

= - 9.94987 \dots

- 9.94987 \dots = - 9.94987 \dots

True

The solutions are

u = 1, u = 199 + 6011, u = 199 - 6011

u = 1, u = 199 + 6011, u = 199 - 6011

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1 : x = 0

e^{x} = 1

Apply exponent rules

e^{x} = 1

f (x) = g (x)

, then

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

ln (e^{x}) = ln (1)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

Solve

e^{x} = 199 + 6011 : x = ln (199 + 6011)

e^{x} = 199 + 6011

Apply exponent rules

e^{x} = 199 + 6011

f (x) = g (x)

, then

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

ln (e^{x}) = ln (199 + 6011)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (199 + 6011)

x = ln (199 + 6011)

Solve

e^{x} = 199 - 6011 : x = ln (199 - 6011)

e^{x} = 199 - 6011

Apply exponent rules

e^{x} = 199 - 6011

f (x) = g (x)

, then

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

ln (e^{x}) = ln (199 - 6011)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (199 - 6011)

x = ln (199 - 6011)

x = 0, x = ln (199 + 6011), x = ln (199 - 6011)

Verify Solutions:

x = 0

False

, x = ln (199 + 6011)

True

, x = ln (199 - 6011)

True

Check the solutions by plugging them into

\frac{\frac{e ^{\frac{x}{2}} - e ^{- \frac{x}{2}}}{2}}{\frac{e ^{x} - e ^{- x}}{2}} = 0.05

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

Plug in

x = 0 :

False

\frac{\frac{e ^{\frac{0}{2}} - e ^{- \frac{0}{2}}}{2}}{\frac{e ^{0} - e ^{- 0}}{2}} = 0.05

Simplify

\frac{\frac{e ^{\frac{0}{2}} - e ^{- \frac{0}{2}}}{2}}{\frac{e ^{0} - e ^{- 0}}{2}} :

Undefined

\frac{\frac{e ^{\frac{0}{2}} - e ^{- \frac{0}{2}}}{2}}{\frac{e ^{0} - e ^{- 0}}{2}}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= \frac{\frac{e ^{0} - e ^{- 0}}{2}}{\frac{e ^{0} - e ^{- 0}}{2}}

\frac{\frac{e ^{0} - e ^{- 0}}{2}}{\frac{e ^{0} - e ^{- 0}}{2}}

is undefined

= Undefined

Undefined

= 0.05

False

Plug in

x = ln (199 + 6011) :

True

\frac{\frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 + 6011)} - e ^{- l n (199 + 6011)}}{2}} = 0.05

\frac{\frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 + 6011)} - e ^{- l n (199 + 6011)}}{2}} = \frac{10 + 3 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

\frac{\frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 + 6011)} - e ^{- l n (199 + 6011)}}{2}}

Divide fractions:

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

= \frac{( e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}} ) \cdot 2}{2 ( e ^{l n (199 + 6011)} - e ^{- l n (199 + 6011)} )}

Cancel the common factor:

2

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{e ^{l n (199 + 6011)} - e ^{- l n (199 + 6011)}}

e^{l n (199 + 6011)} = 199 + 6011

e^{l n (199 + 6011)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 199 + 6011

e^{- l n (199 + 6011)} = (199 + 6011)^{- 1}

e^{- l n (199 + 6011)}

Apply exponent rule:

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

= (e^{l n (199 + 6011)})^{- 1}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199 + 6011)} = 199 + 6011

= (199 + 6011)^{- 1}

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{199 + 60 11 - ( 199 + 60 11 ) ^{- 1}}

Apply exponent rule:

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

(199 + 6011)^{- 1} = \frac{1}{199 + 60 11}

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - e ^{- \frac{l n ( 199 + 60 11 )}{2}}}{199 + 60 11 - \frac{1}{199 + 60 11}}

Apply exponent rule:

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

e^{- \frac{l n ( 199 + 60 11 )}{2}} = \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}}{199 + 60 11 - \frac{1}{199 + 60 11}}

Join

199 + 6011 - \frac{1}{199 + 60 11} : \frac{79200 + 23880 11}{199 + 60 11}

199 + 6011 - \frac{1}{199 + 60 11}

Convert element to fraction:

199 = \frac{199 ( 199 + 60 11 )}{199 + 60 11}, 6011 = \frac{60 \cdot 11 ( 199 + 60 11 )}{199 + 60 11}

= \frac{199 ( 199 + 60 11 )}{199 + 60 11} + \frac{60 11 ( 199 + 60 11 )}{199 + 60 11} - \frac{1}{199 + 60 11}

Since the denominators are equal, combine the fractions:

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

= \frac{199 ( 199 + 60 11 ) + 60 11 ( 199 + 60 11 ) - 1}{199 + 60 11}

Expand

199 (199 + 6011) + 6011 (199 + 6011) - 1 : 79200 + 2388011

199 (199 + 6011) + 6011 (199 + 6011) - 1

Expand

199 (199 + 6011) : 39601 + 1194011

199 (199 + 6011)

Apply the distributive law:

a (b + c) = ab + a c

a = 199, b = 199, c = 6011

= 199 \cdot 199 + 199 \cdot 6011

Simplify

199 \cdot 199 + 199 \cdot 6011 : 39601 + 1194011

199 \cdot 199 + 199 \cdot 6011

Multiply the numbers:

199 \cdot 199 = 39601

= 39601 + 199 \cdot 6011

Multiply the numbers:

199 \cdot 60 = 11940

= 39601 + 1194011

= 39601 + 1194011

= 39601 + 1194011 + 6011 (199 + 6011) - 1

Expand

6011 (199 + 6011) : 1194011 + 39600

6011 (199 + 6011)

Apply the distributive law:

a (b + c) = ab + a c

a = 6011, b = 199, c = 6011

= 6011 \cdot 199 + 6011 \cdot 6011

= 60 \cdot 19911 + 60 \cdot 601111

Simplify

60 \cdot 19911 + 60 \cdot 601111 : 1194011 + 39600

60 \cdot 19911 + 60 \cdot 601111

60 \cdot 19911 = 1194011

60 \cdot 19911

Multiply the numbers:

60 \cdot 199 = 11940

= 1194011

60 \cdot 601111 = 39600

60 \cdot 601111

Multiply the numbers:

60 \cdot 60 = 3600

= 36001111

Apply radical rule:

a a = a

1111 = 11

= 3600 \cdot 11

Multiply the numbers:

3600 \cdot 11 = 39600

= 39600

= 1194011 + 39600

= 1194011 + 39600

= 39601 + 1194011 + 1194011 + 39600 - 1

Simplify

39601 + 1194011 + 1194011 + 39600 - 1 : 79200 + 2388011

39601 + 1194011 + 1194011 + 39600 - 1

Add similar elements:

1194011 + 1194011 = 2388011

= 39601 + 2388011 + 39600 - 1

Add/Subtract the numbers:

39601 + 39600 - 1 = 79200

= 79200 + 2388011

= 79200 + 2388011

= \frac{79200 + 23880 11}{199 + 60 11}

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} - \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}}{\frac{79200 + 23880 11}{199 + 60 11}}

Join

e^{\frac{l n ( 199 + 60 11 )}{2}} - \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}} : \frac{198 + 60 11}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

e^{\frac{l n ( 199 + 60 11 )}{2}} - \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Convert element to fraction:

e^{\frac{l n ( 199 + 60 11 )}{2}} = \frac{e ^{\frac{l n ( 199 + 60 \cdot 11 )}{2}} e ^{\frac{l n ( 199 + 60 11 )}{2}}}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} e ^{\frac{l n ( 199 + 60 11 )}{2}}}{e ^{\frac{l n ( 199 + 60 11 )}{2}}} - \frac{1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{l n ( 199 + 60 11 )}{2}} e ^{\frac{l n ( 199 + 60 11 )}{2}} - 1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

e^{\frac{l n ( 199 + 60 11 )}{2}} e^{\frac{l n ( 199 + 60 11 )}{2}} - 1 = e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}} - 1

e^{\frac{l n ( 199 + 60 11 )}{2}} e^{\frac{l n ( 199 + 60 11 )}{2}} - 1

e^{\frac{l n ( 199 + 60 11 )}{2}} e^{\frac{l n ( 199 + 60 11 )}{2}} = e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}}

e^{\frac{l n ( 199 + 60 11 )}{2}} e^{\frac{l n ( 199 + 60 11 )}{2}}

Apply exponent rule:

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

e^{\frac{l n ( 199 + 60 11 )}{2}} e^{\frac{l n ( 199 + 60 11 )}{2}} = e^{\frac{l n ( 199 + 60 11 )}{2} + \frac{l n ( 199 + 60 11 )}{2}}

= e^{\frac{l n ( 199 + 60 11 )}{2} + \frac{l n ( 199 + 60 11 )}{2}}

Add similar elements:

\frac{ln ( 199 + 60 11 )}{2} + \frac{ln ( 199 + 60 11 )}{2} = 2 \cdot \frac{ln ( 199 + 60 11 )}{2}

= e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}}

= e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}} - 1

= \frac{e ^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}} - 1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}} = 199 + 6011

e^{2 \cdot \frac{l n ( 199 + 60 11 )}{2}}

Multiply

2 \cdot \frac{ln ( 199 + 60 11 )}{2} : ln (199 + 6011)

2 \cdot \frac{ln ( 199 + 60 11 )}{2}

Multiply fractions:

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

= \frac{ln ( 199 + 60 11 ) \cdot 2}{2}

Cancel the common factor:

2

= ln (199 + 6011)

= e^{l n (199 + 6011)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 199 + 6011

= \frac{199 + 60 11 - 1}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Subtract the numbers:

199 - 1 = 198

= \frac{198 + 60 11}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}

= \frac{\frac{198 + 60 11}{e ^{\frac{l n ( 199 + 60 11 )}{2}}}}{\frac{79200 + 23880 11}{199 + 60 11}}

Divide fractions:

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

= \frac{( 198 + 60 11 ) ( 199 + 60 11 )}{e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 79200 + 23880 11 )}

Factor

198 + 6011 : 6 (33 + 1011)

198 + 6011

Rewrite as

= 6 \cdot 33 + 6 \cdot 1011

Factor out common term

6

= 6 (33 + 1011)

= \frac{6 ( 33 + 10 11 ) ( 199 + 60 11 )}{e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 79200 + 23880 11 )}

Factor

79200 + 2388011 : 120 (660 + 19911)

79200 + 2388011

Rewrite as

= 120 \cdot 660 + 120 \cdot 19911

Factor out common term

120

= 120 (660 + 19911)

= \frac{6 ( 33 + 10 11 ) ( 199 + 60 11 )}{e ^{\frac{l n ( 199 + 60 11 )}{2}} \cdot 120 ( 660 + 199 11 )}

Cancel the common factor:

6

= \frac{( 33 + 10 11 ) ( 199 + 60 11 )}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 660 + 199 11 )}

Expand

(33 + 1011) (199 + 6011) : 13167 + 397011

(33 + 1011) (199 + 6011)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 33, b = 1011, c = 199, d = 6011

= 33 \cdot 199 + 33 \cdot 6011 + 1011 \cdot 199 + 1011 \cdot 6011

= 33 \cdot 199 + 33 \cdot 6011 + 10 \cdot 19911 + 10 \cdot 601111

Simplify

33 \cdot 199 + 33 \cdot 6011 + 10 \cdot 19911 + 10 \cdot 601111 : 13167 + 397011

33 \cdot 199 + 33 \cdot 6011 + 10 \cdot 19911 + 10 \cdot 601111

33 \cdot 199 = 6567

33 \cdot 199

Multiply the numbers:

33 \cdot 199 = 6567

= 6567

33 \cdot 6011 = 198011

33 \cdot 6011

Multiply the numbers:

33 \cdot 60 = 1980

= 198011

10 \cdot 19911 = 199011

10 \cdot 19911

Multiply the numbers:

10 \cdot 199 = 1990

= 199011

10 \cdot 601111 = 6600

10 \cdot 601111

Multiply the numbers:

10 \cdot 60 = 600

= 6001111

Apply radical rule:

a a = a

1111 = 11

= 600 \cdot 11

Multiply the numbers:

600 \cdot 11 = 6600

= 6600

= 6567 + 198011 + 199011 + 6600

Add similar elements:

198011 + 199011 = 397011

= 6567 + 397011 + 6600

Add the numbers:

6567 + 6600 = 13167

= 13167 + 397011

= 13167 + 397011

= \frac{13167 + 3970 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 660 + 199 11 )}

Rationalize

\frac{13167 + 3970 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 660 + 199 11 )} : \frac{10 + 3 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

\frac{13167 + 3970 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 660 + 199 11 )}

Multiply by the conjugate

\frac{660 - 199 11}{660 - 199 11}

= \frac{( 13167 + 3970 11 ) ( 660 - 199 11 )}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}} ( 660 + 199 11 ) ( 660 - 199 11 )}

(13167 + 397011) (660 - 19911) = - 110 - 3311

(13167 + 397011) (660 - 19911)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 13167, b = 397011, c = 660, d = - 19911

= 13167 \cdot 660 + 13167 (- 19911) + 397011 \cdot 660 + 397011 (- 19911)

Apply minus-plus rules

+ (- a) = - a

= 13167 \cdot 660 - 13167 \cdot 19911 + 3970 \cdot 66011 - 3970 \cdot 1991111

Simplify

13167 \cdot 660 - 13167 \cdot 19911 + 3970 \cdot 66011 - 3970 \cdot 1991111 : - 110 - 3311

13167 \cdot 660 - 13167 \cdot 19911 + 3970 \cdot 66011 - 3970 \cdot 1991111

13167 \cdot 660 = 8690220

13167 \cdot 660

Multiply the numbers:

13167 \cdot 660 = 8690220

= 8690220

13167 \cdot 19911 = 262023311

13167 \cdot 19911

Multiply the numbers:

13167 \cdot 199 = 2620233

= 262023311

3970 \cdot 66011 = 262020011

3970 \cdot 66011

Multiply the numbers:

3970 \cdot 660 = 2620200

= 262020011

3970 \cdot 1991111 = 8690330

3970 \cdot 1991111

Multiply the numbers:

3970 \cdot 199 = 790030

= 7900301111

Apply radical rule:

a a = a

1111 = 11

= 790030 \cdot 11

Multiply the numbers:

790030 \cdot 11 = 8690330

= 8690330

= 8690220 - 262023311 + 262020011 - 8690330

Add similar elements:

- 262023311 + 262020011 = - 3311

= 8690220 - 3311 - 8690330

Subtract the numbers:

8690220 - 8690330 = - 110

= - 110 - 3311

= - 110 - 3311

20 e^{\frac{l n ( 199 + 60 11 )}{2}} (660 + 19911) (660 - 19911) = - 220 e^{\frac{l n ( 199 + 60 11 )}{2}}

20 e^{\frac{l n ( 199 + 60 11 )}{2}} (660 + 19911) (660 - 19911)

Expand

(660 + 19911) (660 - 19911) : - 11

(660 + 19911) (660 - 19911)

Apply Difference of Two Squares Formula:

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

a = 660, b = 19911

= 66 0^{2} - (19911)^{2}

Simplify

66 0^{2} - (19911)^{2} : - 11

66 0^{2} - (19911)^{2}

66 0^{2} = 435600

66 0^{2}

66 0^{2} = 435600

= 435600

(19911)^{2} = 435611

(19911)^{2}

Apply exponent rule:

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

= 19 9^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 19 9^{2} \cdot 11

19 9^{2} = 39601

= 39601 \cdot 11

Multiply the numbers:

39601 \cdot 11 = 435611

= 435611

= 435600 - 435611

Subtract the numbers:

435600 - 435611 = - 11

= - 11

= - 11

= 20 e^{\frac{l n ( 199 + 60 11 )}{2}} (- 11)

Expand

20 e^{\frac{l n ( 199 + 60 11 )}{2}} (- 11) : - 220 e^{\frac{l n ( 199 + 60 11 )}{2}}

20 e^{\frac{l n ( 199 + 60 11 )}{2}} (- 11)

Distribute parentheses

= 20 e^{\frac{l n ( 199 + 60 11 )}{2}} (- 11)

Apply minus-plus rules

+ (- a) = - a

= - 20 \cdot 11 e^{\frac{l n ( 199 + 60 11 )}{2}}

Multiply the numbers:

20 \cdot 11 = 220

= - 220 e^{\frac{l n ( 199 + 60 11 )}{2}}

= - 220 e^{\frac{l n ( 199 + 60 11 )}{2}}

= \frac{- 110 - 33 11}{- 220 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Apply the fraction rule:

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

- 110 - 3311 = - (110 + 3311)

= \frac{110 + 33 11}{220 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Factor

110 + 3311 : 11 (10 + 311)

110 + 3311

Rewrite as

= 11 \cdot 10 + 11 \cdot 311

Factor out common term

11

= 11 (10 + 311)

= \frac{11 ( 10 + 3 11 )}{220 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

Cancel the common factor:

11

= \frac{10 + 3 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

= \frac{10 + 3 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}}}

\frac{10 + 3 11}{20 e ^{\frac{l n ( 199 + 60 11 )}{2}}} = 0.05

True

Plug in

x = ln (199 - 6011) :

True

\frac{\frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)}}{2}} = 0.05

\frac{\frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)}}{2}} = \frac{10 - 3 11}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

\frac{\frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}}}{2}}{\frac{e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)}}{2}}

Divide fractions:

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

= \frac{( e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}} ) \cdot 2}{2 ( e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)} )}

Cancel the common factor:

2

= \frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}}}{e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)}}

Apply the fraction rule:

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

\frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} - e ^{- \frac{l n ( 199 - 60 11 )}{2}}}{e ^{l n (199 - 6011)} - e ^{- l n (199 - 6011)}} = \frac{- ( e ^{- \frac{l n ( 199 - 60 11 )}{2}} - e ^{\frac{l n ( 199 - 60 11 )}{2}} )}{- ( e ^{- l n (199 - 6011)} - e ^{l n (199 - 6011)} )}

= \frac{e ^{- \frac{l n ( 199 - 60 11 )}{2}} - e ^{\frac{l n ( 199 - 60 11 )}{2}}}{e ^{- l n (199 - 6011)} - e ^{l n (199 - 6011)}}

e^{- l n (199 - 6011)} = (199 - 6011)^{- 1}

e^{- l n (199 - 6011)}

Apply exponent rule:

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

= (e^{l n (199 - 6011)})^{- 1}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (199 - 6011)} = 199 - 6011

= (199 - 6011)^{- 1}

e^{l n (199 - 6011)} = 199 - 6011

e^{l n (199 - 6011)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 199 - 6011

= \frac{e ^{- \frac{l n ( 199 - 60 11 )}{2}} - e ^{\frac{l n ( 199 - 60 11 )}{2}}}{( 199 - 60 11 ) ^{- 1} - ( 199 - 60 11 )}

Apply exponent rule:

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

(199 - 6011)^{- 1} = \frac{1}{199 - 60 11}

= \frac{e ^{- \frac{l n ( 199 - 60 11 )}{2}} - e ^{\frac{l n ( 199 - 60 11 )}{2}}}{\frac{1}{199 - 60 11} - ( 199 - 60 11 )}

Apply exponent rule:

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

e^{- \frac{l n ( 199 - 60 11 )}{2}} = \frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

= \frac{\frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}} - e ^{\frac{l n ( 199 - 60 11 )}{2}}}{\frac{1}{199 - 60 11} - ( 199 - 60 11 )}

Join

\frac{1}{199 - 60 11} - (199 - 6011) : \frac{23880 11 - 79200}{199 - 60 11}

\frac{1}{199 - 60 11} - (199 - 6011)

Convert element to fraction:

(199 - 6011) = \frac{( 199 - 60 \cdot 11 ) ( 199 - 60 11 )}{199 - 60 11}

= \frac{1}{199 - 60 11} - \frac{( 199 - 60 11 ) ( 199 - 60 11 )}{199 - 60 11}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - ( 199 - 60 11 ) ( 199 - 60 11 )}{199 - 60 11}

1 - (199 - 6011) (199 - 6011) = 1 - (199 - 6011)^{2}

1 - (199 - 6011) (199 - 6011)

(199 - 6011) (199 - 6011) = (199 - 6011)^{2}

(199 - 6011) (199 - 6011)

Apply exponent rule:

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

(199 - 6011) (199 - 6011) = (199 - 6011)^{1 + 1}

= (199 - 6011)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (199 - 6011)^{2}

= 1 - (199 - 6011)^{2}

= \frac{1 - ( 199 - 60 11 ) ^{2}}{199 - 60 11}

Expand

1 - (199 - 6011)^{2} : 2388011 - 79200

1 - (199 - 6011)^{2}

(199 - 6011)^{2} : 79201 - 2388011

Apply Perfect Square Formula:

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

a = 199, b = 6011

= 19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2}

Simplify

19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2} : 79201 - 2388011

19 9^{2} - 2 \cdot 199 \cdot 6011 + (6011)^{2}

19 9^{2} = 39601

19 9^{2}

19 9^{2} = 39601

= 39601

2 \cdot 199 \cdot 6011 = 2388011

2 \cdot 199 \cdot 6011

Multiply the numbers:

2 \cdot 199 \cdot 60 = 23880

= 2388011

(6011)^{2} = 39600

(6011)^{2}

Apply exponent rule:

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

= 6 0^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 6 0^{2} \cdot 11

6 0^{2} = 3600

= 3600 \cdot 11

Multiply the numbers:

3600 \cdot 11 = 39600

= 39600

= 39601 - 2388011 + 39600

Add the numbers:

39601 + 39600 = 79201

= 79201 - 2388011

= 79201 - 2388011

= 1 - (79201 - 2388011)

- (79201 - 2388011) : - 79201 + 2388011

- (79201 - 2388011)

Distribute parentheses

= - (79201) - (- 2388011)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 79201 + 2388011

= 1 - 79201 + 2388011

Subtract the numbers:

1 - 79201 = - 79200

= 2388011 - 79200

= \frac{23880 11 - 79200}{199 - 60 11}

= \frac{\frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}} - e ^{\frac{l n ( 199 - 60 11 )}{2}}}{\frac{23880 11 - 79200}{199 - 60 11}}

Join

\frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}} - e^{\frac{l n ( 199 - 60 11 )}{2}} : \frac{60 11 - 198}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

\frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}} - e^{\frac{l n ( 199 - 60 11 )}{2}}

Convert element to fraction:

e^{\frac{l n ( 199 - 60 11 )}{2}} = \frac{e ^{\frac{l n ( 199 - 60 \cdot 11 )}{2}} e ^{\frac{l n ( 199 - 60 11 )}{2}}}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

= \frac{1}{e ^{\frac{l n ( 199 - 60 11 )}{2}}} - \frac{e ^{\frac{l n ( 199 - 60 11 )}{2}} e ^{\frac{l n ( 199 - 60 11 )}{2}}}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - e ^{\frac{l n ( 199 - 60 11 )}{2}} e ^{\frac{l n ( 199 - 60 11 )}{2}}}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

1 - e^{\frac{l n ( 199 - 60 11 )}{2}} e^{\frac{l n ( 199 - 60 11 )}{2}} = 1 - e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}

1 - e^{\frac{l n ( 199 - 60 11 )}{2}} e^{\frac{l n ( 199 - 60 11 )}{2}}

e^{\frac{l n ( 199 - 60 11 )}{2}} e^{\frac{l n ( 199 - 60 11 )}{2}} = e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}

e^{\frac{l n ( 199 - 60 11 )}{2}} e^{\frac{l n ( 199 - 60 11 )}{2}}

Apply exponent rule:

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

e^{\frac{l n ( 199 - 60 11 )}{2}} e^{\frac{l n ( 199 - 60 11 )}{2}} = e^{\frac{l n ( 199 - 60 11 )}{2} + \frac{l n ( 199 - 60 11 )}{2}}

= e^{\frac{l n ( 199 - 60 11 )}{2} + \frac{l n ( 199 - 60 11 )}{2}}

Add similar elements:

\frac{ln ( 199 - 60 11 )}{2} + \frac{ln ( 199 - 60 11 )}{2} = 2 \cdot \frac{ln ( 199 - 60 11 )}{2}

= e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}

= 1 - e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}

= \frac{1 - e ^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}} = 199 - 6011

e^{2 \cdot \frac{l n ( 199 - 60 11 )}{2}}

Multiply

2 \cdot \frac{ln ( 199 - 60 11 )}{2} : ln (199 - 6011)

2 \cdot \frac{ln ( 199 - 60 11 )}{2}

Multiply fractions:

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

= \frac{ln ( 199 - 60 11 ) \cdot 2}{2}

Cancel the common factor:

2

= ln (199 - 6011)

= e^{l n (199 - 6011)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 199 - 6011

= \frac{1 - ( 199 - 60 11 )}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

Expand

1 - (199 - 6011) : 6011 - 198

1 - (199 - 6011)

- (199 - 6011) : - 199 + 6011

- (199 - 6011)

Distribute parentheses

= - (199) - (- 6011)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 199 + 6011

= 1 - 199 + 6011

Subtract the numbers:

1 - 199 = - 198

= 6011 - 198

= \frac{60 11 - 198}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}

= \frac{\frac{60 11 - 198}{e ^{\frac{l n ( 199 - 60 11 )}{2}}}}{\frac{23880 11 - 79200}{199 - 60 11}}

Divide fractions:

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

= \frac{( 60 11 - 198 ) ( 199 - 60 11 )}{e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 23880 11 - 79200 )}

Factor

6011 - 198 : 6 (1011 - 33)

6011 - 198

Rewrite as

= 6 \cdot 1011 - 6 \cdot 33

Factor out common term

6

= 6 (1011 - 33)

= \frac{6 ( 10 11 - 33 ) ( 199 - 60 11 )}{e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 23880 11 - 79200 )}

Factor

2388011 - 79200 : 120 (19911 - 660)

2388011 - 79200

Rewrite as

= 120 \cdot 19911 - 120 \cdot 660

Factor out common term

120

= 120 (19911 - 660)

= \frac{6 ( 10 11 - 33 ) ( 199 - 60 11 )}{e ^{\frac{l n ( 199 - 60 11 )}{2}} \cdot 120 ( 199 11 - 660 )}

Cancel the common factor:

6

= \frac{( 10 11 - 33 ) ( 199 - 60 11 )}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 199 11 - 660 )}

Expand

(1011 - 33) (199 - 6011) : 397011 - 13167

(1011 - 33) (199 - 6011)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 1011, b = - 33, c = 199, d = - 6011

= 1011 \cdot 199 + 1011 (- 6011) + (- 33) \cdot 199 + (- 33) (- 6011)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 10 \cdot 19911 - 10 \cdot 601111 - 33 \cdot 199 + 33 \cdot 6011

Simplify

10 \cdot 19911 - 10 \cdot 601111 - 33 \cdot 199 + 33 \cdot 6011 : 397011 - 13167

10 \cdot 19911 - 10 \cdot 601111 - 33 \cdot 199 + 33 \cdot 6011

10 \cdot 19911 = 199011

10 \cdot 19911

Multiply the numbers:

10 \cdot 199 = 1990

= 199011

10 \cdot 601111 = 6600

10 \cdot 601111

Multiply the numbers:

10 \cdot 60 = 600

= 6001111

Apply radical rule:

a a = a

1111 = 11

= 600 \cdot 11

Multiply the numbers:

600 \cdot 11 = 6600

= 6600

33 \cdot 199 = 6567

33 \cdot 199

Multiply the numbers:

33 \cdot 199 = 6567

= 6567

33 \cdot 6011 = 198011

33 \cdot 6011

Multiply the numbers:

33 \cdot 60 = 1980

= 198011

= 199011 - 6600 - 6567 + 198011

Add similar elements:

199011 + 198011 = 397011

= 397011 - 6600 - 6567

Subtract the numbers:

- 6600 - 6567 = - 13167

= 397011 - 13167

= 397011 - 13167

= \frac{3970 11 - 13167}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 199 11 - 660 )}

Rationalize

\frac{3970 11 - 13167}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 199 11 - 660 )} : \frac{10 - 3 11}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

\frac{3970 11 - 13167}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 199 11 - 660 )}

Multiply by the conjugate

\frac{199 11 + 660}{199 11 + 660}

= \frac{( 3970 11 - 13167 ) ( 199 11 + 660 )}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}} ( 199 11 - 660 ) ( 199 11 + 660 )}

(397011 - 13167) (19911 + 660) = 110 - 3311

(397011 - 13167) (19911 + 660)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 397011, b = - 13167, c = 19911, d = 660

= 397011 \cdot 19911 + 397011 \cdot 660 + (- 13167) \cdot 19911 + (- 13167) \cdot 660

Apply minus-plus rules

+ (- a) = - a

= 3970 \cdot 1991111 + 3970 \cdot 66011 - 13167 \cdot 19911 - 13167 \cdot 660

Simplify

3970 \cdot 1991111 + 3970 \cdot 66011 - 13167 \cdot 19911 - 13167 \cdot 660 : 110 - 3311

3970 \cdot 1991111 + 3970 \cdot 66011 - 13167 \cdot 19911 - 13167 \cdot 660

3970 \cdot 1991111 = 8690330

3970 \cdot 1991111

Multiply the numbers:

3970 \cdot 199 = 790030

= 7900301111

Apply radical rule:

a a = a

1111 = 11

= 790030 \cdot 11

Multiply the numbers:

790030 \cdot 11 = 8690330

= 8690330

3970 \cdot 66011 = 262020011

3970 \cdot 66011

Multiply the numbers:

3970 \cdot 660 = 2620200

= 262020011

13167 \cdot 19911 = 262023311

13167 \cdot 19911

Multiply the numbers:

13167 \cdot 199 = 2620233

= 262023311

13167 \cdot 660 = 8690220

13167 \cdot 660

Multiply the numbers:

13167 \cdot 660 = 8690220

= 8690220

= 8690330 + 262020011 - 262023311 - 8690220

Add similar elements:

262020011 - 262023311 = - 3311

= 8690330 - 3311 - 8690220

Subtract the numbers:

8690330 - 8690220 = 110

= 110 - 3311

= 110 - 3311

20 e^{\frac{l n ( 199 - 60 11 )}{2}} (19911 - 660) (19911 + 660) = 220 e^{\frac{l n ( 199 - 60 11 )}{2}}

20 e^{\frac{l n ( 199 - 60 11 )}{2}} (19911 - 660) (19911 + 660)

Expand

(19911 - 660) (19911 + 660) : 11

(19911 - 660) (19911 + 660)

Apply Difference of Two Squares Formula:

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

a = 19911, b = 660

= (19911)^{2} - 66 0^{2}

Simplify

(19911)^{2} - 66 0^{2} : 11

(19911)^{2} - 66 0^{2}

(19911)^{2} = 435611

(19911)^{2}

Apply exponent rule:

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

= 19 9^{2} (11)^{2}

(11)^{2} : 11

Apply radical rule:

a = a^{\frac{1}{2}}

= (1 1^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 1 1^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 11

= 19 9^{2} \cdot 11

19 9^{2} = 39601

= 39601 \cdot 11

Multiply the numbers:

39601 \cdot 11 = 435611

= 435611

66 0^{2} = 435600

66 0^{2}

66 0^{2} = 435600

= 435600

= 435611 - 435600

Subtract the numbers:

435611 - 435600 = 11

= 11

= 11

= 20 e^{\frac{l n ( 199 - 60 11 )}{2}} \cdot 11

Expand

20 e^{\frac{l n ( 199 - 60 11 )}{2}} \cdot 11 : 220 e^{\frac{l n ( 199 - 60 11 )}{2}}

20 e^{\frac{l n ( 199 - 60 11 )}{2}} \cdot 11

Distribute parentheses

= 20 e^{\frac{l n ( 199 - 60 11 )}{2}} \cdot 11

= 20 \cdot 11 e^{\frac{l n ( 199 - 60 11 )}{2}}

Multiply the numbers:

20 \cdot 11 = 220

= 220 e^{\frac{l n ( 199 - 60 11 )}{2}}

= 220 e^{\frac{l n ( 199 - 60 11 )}{2}}

= \frac{110 - 33 11}{220 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

Factor

110 - 3311 : 11 (10 - 311)

110 - 3311

Rewrite as

= 11 \cdot 10 - 11 \cdot 311

Factor out common term

11

= 11 (10 - 311)

= \frac{11 ( 10 - 3 11 )}{220 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

Cancel the common factor:

11

= \frac{10 - 3 11}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

= \frac{10 - 3 11}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}}}

\frac{10 - 3 11}{20 e ^{\frac{l n ( 199 - 60 11 )}{2}}} = 0.05

True

The solutions are

x = ln (199 + 6011), x = ln (199 - 6011)

x = ln (199 + 6011), x = ln (199 - 6011)

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Frequently Asked Questions (FAQ)

What is the general solution for 0.05=(sinh(x/2))/(sinh(x)) ?
The general solution for 0.05=(sinh(x/2))/(sinh(x)) is x=ln(199+60sqrt(11)),x=ln(199-60sqrt(11))