What is the general solution for (cosh(0.2m))/(cosh(0.4m))=0.6 ?

The general solution for (cosh(0.2m))/(cosh(0.4m))=0.6 is m=ln(0.03402…),m=ln(29.38731…)

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(cosh(0.2m))/(cosh(0.4m))=0.6

Solution

\frac{cosh ( 0.2 m )}{cosh ( 0.4 m )} = 0.6

Solution

m = ln (0.03402 \dots), m = ln (29.38731 \dots)

Degrees

m = - 193.69199 \dots^{\circ}, m = 193.69199 \dots^{\circ}

Solution steps

\frac{cosh ( 0.2 m )}{cosh ( 0.4 m )} = 0.6

Rewrite using trig identities

\frac{cosh ( 0.2 m )}{cosh ( 0.4 m )} = 0.6

Use the Hyperbolic identity:

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

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} = 0.6

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} = 0.6

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} = 0.6 : m = ln (0.03402 \dots), m = ln (29.38731 \dots)

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} = 0.6

Multiply both sides by

\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} \cdot \frac{e ^{0.4 m} + e ^{- 0.4 m}}{2} = 0.6 \cdot \frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}

Simplify

\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2} = \frac{0.6 ( e ^{0.4 m} + e ^{- 0.4 m} )}{2}

Apply exponent rules

\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2} = \frac{0.6 ( e ^{0.4 m} + e ^{- 0.4 m} )}{2}

Apply exponent rule:

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

e^{0.2 m} = (e^{m})^{0.2}, e^{- 0.2 m} = (e^{m})^{- 0.2}, e^{0.4 m} = (e^{m})^{0.4}, e^{- 0.4 m} = (e^{m})^{- 0.4}

\frac{( e ^{m} ) ^{0.2} + ( e ^{m} ) ^{- 0.2}}{2} = \frac{0.6 ( ( e ^{m} ) ^{0.4} + ( e ^{m} ) ^{- 0.4} )}{2}

\frac{( e ^{m} ) ^{0.2} + ( e ^{m} ) ^{- 0.2}}{2} = \frac{0.6 ( ( e ^{m} ) ^{0.4} + ( e ^{m} ) ^{- 0.4} )}{2}

Rewrite the equation with

e^{m} = u

\frac{( u ) ^{0.2} + ( u ) ^{- 0.2}}{2} = \frac{0.6 ( ( u ) ^{0.4} + ( u ) ^{- 0.4} )}{2}

Solve

\frac{u ^{0.2} + u ^{- 0.2}}{2} = \frac{0.6 ( u ^{0.4} + u ^{- 0.4} )}{2} : u = 0.03402 \dots, u = 29.38731 \dots

\frac{u ^{0.2} + u ^{- 0.2}}{2} = \frac{0.6 ( u ^{0.4} + u ^{- 0.4} )}{2}

Multiply both sides by

2

\frac{u ^{0.2} + u ^{- 0.2}}{2} \cdot 2 = \frac{0.6 ( u ^{0.4} + u ^{- 0.4} )}{2} \cdot 2

Simplify

u^{0.2} + u^{- 0.2} = 0.6 (u^{0.4} + u^{- 0.4})

Expand

u^{0.2} + u^{- 0.2} : u^{0.2} + \frac{1}{u ^{0.2}}

u^{0.2} + u^{- 0.2}

Apply exponent rule:

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

= u^{0.2} + \frac{1}{u ^{0.2}}

Expand

0.6 (u^{0.4} + u^{- 0.4}) : 0.6 u^{0.4} + \frac{0.6}{u ^{0.4}}

0.6 (u^{0.4} + u^{- 0.4})

Apply exponent rule:

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

= 0.6 (u^{0.4} + \frac{1}{u ^{0.4}})

Apply the distributive law:

a (b + c) = ab + a c

a = 0.6, b = u^{0.4}, c = \frac{1}{u ^{0.4}}

= 0.6 u^{0.4} + 0.6 \cdot \frac{1}{u ^{0.4}}

0.6 \cdot \frac{1}{u ^{0.4}} = \frac{0.6}{u ^{0.4}}

0.6 \cdot \frac{1}{u ^{0.4}}

Multiply fractions:

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

= \frac{1 \cdot 0.6}{u ^{0.4}}

Multiply the numbers:

1 \cdot 0.6 = 0.6

= \frac{0.6}{u ^{0.4}}

= 0.6 u^{0.4} + \frac{0.6}{u ^{0.4}}

u^{0.2} + \frac{1}{u ^{0.2}} = 0.6 u^{0.4} + \frac{0.6}{u ^{0.4}}

Use the following exponent property

Rewrite the equation with

v + \frac{1}{v} = 0.6 v^{2} + \frac{0.6}{v ^{2}}

Solve

v + \frac{1}{v} = 0.6 v^{2} + \frac{0.6}{v ^{2}} : v \approx 0.50859 \dots, v \approx 1.96621 \dots

v + \frac{1}{v} = 0.6 v^{2} + \frac{0.6}{v ^{2}}

Multiply by LCM

v + \frac{1}{v} = 0.6 v^{2} + \frac{0.6}{v ^{2}}

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

v \cdot 10 + \frac{1}{v} \cdot 10 = 0.6 v^{2} \cdot 10 + \frac{0.6}{v ^{2}} \cdot 10

Refine

10 v + \frac{10}{v} = 6 v^{2} + \frac{6}{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}

10 v v^{2} + \frac{10}{v} v^{2} = 6 v^{2} v^{2} + \frac{6}{v ^{2}} v^{2}

Simplify

10 v v^{2} + \frac{10}{v} v^{2} = 6 v^{2} v^{2} + \frac{6}{v ^{2}} v^{2}

Simplify

10 v v^{2} : 10 v^{3}

10 v v^{2}

Apply exponent rule:

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

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

= 10 v^{1 + 2}

Add the numbers:

1 + 2 = 3

= 10 v^{3}

Simplify

\frac{10}{v} v^{2} : 10 v

\frac{10}{v} v^{2}

Multiply fractions:

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

= \frac{10 v ^{2}}{v}

Cancel the common factor:

v

= 10 v

Simplify

6 v^{2} v^{2} : 6 v^{4}

6 v^{2} v^{2}

Apply exponent rule:

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

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

= 6 v^{2 + 2}

Add the numbers:

2 + 2 = 4

= 6 v^{4}

Simplify

\frac{6}{v ^{2}} v^{2} : 6

\frac{6}{v ^{2}} v^{2}

Multiply fractions:

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

= \frac{6 v ^{2}}{v ^{2}}

Cancel the common factor:

v^{2}

= 6

10 v^{3} + 10 v = 6 v^{4} + 6

10 v^{3} + 10 v = 6 v^{4} + 6

10 v^{3} + 10 v = 6 v^{4} + 6

Solve

10 v^{3} + 10 v = 6 v^{4} + 6 : v \approx 0.50859 \dots, v \approx 1.96621 \dots

10 v^{3} + 10 v = 6 v^{4} + 6

Switch sides

6 v^{4} + 6 = 10 v^{3} + 10 v

Move

10 v

to the left side

6 v^{4} + 6 = 10 v^{3} + 10 v

Subtract

10 v

from both sides

6 v^{4} + 6 - 10 v = 10 v^{3} + 10 v - 10 v

Simplify

6 v^{4} + 6 - 10 v = 10 v^{3}

6 v^{4} + 6 - 10 v = 10 v^{3}

Move

10 v^{3}

to the left side

6 v^{4} + 6 - 10 v = 10 v^{3}

Subtract

10 v^{3}

from both sides

6 v^{4} + 6 - 10 v - 10 v^{3} = 10 v^{3} - 10 v^{3}

Simplify

6 v^{4} + 6 - 10 v - 10 v^{3} = 0

6 v^{4} + 6 - 10 v - 10 v^{3} = 0

Write in the standard form

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

6 v^{4} - 10 v^{3} - 10 v + 6 = 0

Find one solution for

6 v^{4} - 10 v^{3} - 10 v + 6 = 0

using Newton-Raphson:

v \approx 0.50859 \dots

6 v^{4} - 10 v^{3} - 10 v + 6 = 0

Newton-Raphson Approximation Definition

f (v) = 6 v^{4} - 10 v^{3} - 10 v + 6

Find

f^{^{'}} (v) : 24 v^{3} - 30 v^{2} - 10

\frac{d}{d v} (6 v^{4} - 10 v^{3} - 10 v + 6)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (6 v^{4}) - \frac{d}{d v} (10 v^{3}) - \frac{d}{d v} (10 v) + \frac{d}{d v} (6)

\frac{d}{d v} (6 v^{4}) = 24 v^{3}

\frac{d}{d v} (6 v^{4})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d}{d v} (v^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 \cdot 4 v^{4 - 1}

Simplify

= 24 v^{3}

\frac{d}{d v} (10 v^{3}) = 30 v^{2}

\frac{d}{d v} (10 v^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d}{d v} (v^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 10 \cdot 3 v^{3 - 1}

Simplify

= 30 v^{2}

\frac{d}{d v} (10 v) = 10

\frac{d}{d v} (10 v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 10 \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 10 \cdot 1

Simplify

= 10

\frac{d}{d v} (6) = 0

\frac{d}{d v} (6)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 24 v^{3} - 30 v^{2} - 10 + 0

Simplify

= 24 v^{3} - 30 v^{2} - 10

Let

v_{0} = 1

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0.5 : Δ v_{1} = 0.5

f (v_{0}) = 6 \cdot 1^{4} - 10 \cdot 1^{3} - 10 \cdot 1 + 6 = - 8

f^{^{'}} (v_{0}) = 24 \cdot 1^{3} - 30 \cdot 1^{2} - 10 = - 16

v_{1} = 0.5

Δ v_{1} = ∣ 0.5 - 1 ∣ = 0.5

Δ v_{1} = 0.5

v_{2} = 0.50862 \dots : Δ v_{2} = 0.00862 \dots

f (v_{1}) = 6 \cdot 0. 5^{4} - 10 \cdot 0. 5^{3} - 10 \cdot 0.5 + 6 = 0.125

f^{^{'}} (v_{1}) = 24 \cdot 0. 5^{3} - 30 \cdot 0. 5^{2} - 10 = - 14.5

v_{2} = 0.50862 \dots

Δ v_{2} = ∣ 0.50862 \dots - 0.5 ∣ = 0.00862 \dots

Δ v_{2} = 0.00862 \dots

v_{3} = 0.50859 \dots : Δ v_{3} = 0.00003 \dots

f (v_{2}) = 6 \cdot 0.50862 \dots^{4} - 10 \cdot 0.50862 \dots^{3} - 10 \cdot 0.50862 \dots + 6 = - 0.00044 \dots

f^{^{'}} (v_{2}) = 24 \cdot 0.50862 \dots^{3} - 30 \cdot 0.50862 \dots^{2} - 10 = - 14.60298 \dots

v_{3} = 0.50859 \dots

Δ v_{3} = ∣ 0.50859 \dots - 0.50862 \dots ∣ = 0.00003 \dots

Δ v_{3} = 0.00003 \dots

v_{4} = 0.50859 \dots : Δ v_{4} = 3.77392 E - 10

f (v_{3}) = 6 \cdot 0.50859 \dots^{4} - 10 \cdot 0.50859 \dots^{3} - 10 \cdot 0.50859 \dots + 6 = - 5.51091 E - 9

f^{^{'}} (v_{3}) = 24 \cdot 0.50859 \dots^{3} - 30 \cdot 0.50859 \dots^{2} - 10 = - 14.60262 \dots

v_{4} = 0.50859 \dots

Δ v_{4} = ∣ 0.50859 \dots - 0.50859 \dots ∣ = 3.77392 E - 10

Δ v_{4} = 3.77392 E - 10

v \approx 0.50859 \dots

Apply long division:

\frac{6 v ^{4} - 10 v ^{3} - 10 v + 6}{v - 0.50859 \dots} = 6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots

6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots \approx 0

Find one solution for

6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots = 0

using Newton-Raphson:

v \approx 1.96621 \dots

6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots = 0

Newton-Raphson Approximation Definition

f (v) = 6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots

Find

f^{^{'}} (v) : 18 v^{2} - 13.89691 \dots v - 3.53391 \dots

\frac{d}{d v} (6 v^{3} - 6.94845 \dots v^{2} - 3.53391 \dots v - 11.79731 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (6 v^{3}) - \frac{d}{d v} (6.94845 \dots v^{2}) - \frac{d}{d v} (3.53391 \dots v) - \frac{d}{d v} (11.79731 \dots)

\frac{d}{d v} (6 v^{3}) = 18 v^{2}

\frac{d}{d v} (6 v^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d}{d v} (v^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 \cdot 3 v^{3 - 1}

Simplify

= 18 v^{2}

\frac{d}{d v} (6.94845 \dots v^{2}) = 13.89691 \dots v

\frac{d}{d v} (6.94845 \dots v^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6.94845 \dots \frac{d}{d v} (v^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6.94845 \dots \cdot 2 v^{2 - 1}

Simplify

= 13.89691 \dots v

\frac{d}{d v} (3.53391 \dots v) = 3.53391 \dots

\frac{d}{d v} (3.53391 \dots v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 3.53391 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 3.53391 \dots \cdot 1

Simplify

= 3.53391 \dots

\frac{d}{d v} (11.79731 \dots) = 0

\frac{d}{d v} (11.79731 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 18 v^{2} - 13.89691 \dots v - 3.53391 \dots - 0

Simplify

= 18 v^{2} - 13.89691 \dots v - 3.53391 \dots

Let

v_{0} = - 3

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = - 1.87222 \dots : Δ v_{1} = 1.12777 \dots

f (v_{0}) = 6 (- 3)^{3} - 6.94845 \dots (- 3)^{2} - 3.53391 \dots (- 3) - 11.79731 \dots = - 225.73168 \dots

f^{^{'}} (v_{0}) = 18 (- 3)^{2} - 13.89691 \dots (- 3) - 3.53391 \dots = 200.15683 \dots

v_{1} = - 1.87222 \dots

Δ v_{1} = ∣ - 1.87222 \dots - (- 3) ∣ = 1.12777 \dots

Δ v_{1} = 1.12777 \dots

v_{2} = - 1.06697 \dots : Δ v_{2} = 0.80525 \dots

f (v_{1}) = 6 (- 1.87222 \dots)^{3} - 6.94845 \dots (- 1.87222 \dots)^{2} - 3.53391 \dots (- 1.87222 \dots) - 11.79731 \dots = - 68.91246 \dots

f^{^{'}} (v_{1}) = 18 (- 1.87222 \dots)^{2} - 13.89691 \dots (- 1.87222 \dots) - 3.53391 \dots = 85.57838 \dots

v_{2} = - 1.06697 \dots

Δ v_{2} = ∣ - 1.06697 \dots - (- 1.87222 \dots) ∣ = 0.80525 \dots

Δ v_{2} = 0.80525 \dots

v_{3} = - 0.33628 \dots : Δ v_{3} = 0.73068 \dots

f (v_{2}) = 6 (- 1.06697 \dots)^{3} - 6.94845 \dots (- 1.06697 \dots)^{2} - 3.53391 \dots (- 1.06697 \dots) - 11.79731 \dots = - 23.22503 \dots

f^{^{'}} (v_{2}) = 18 (- 1.06697 \dots)^{2} - 13.89691 \dots (- 1.06697 \dots) - 3.53391 \dots = 31.78535 \dots

v_{3} = - 0.33628 \dots

Δ v_{3} = ∣ - 0.33628 \dots - (- 1.06697 \dots) ∣ = 0.73068 \dots

Δ v_{3} = 0.73068 \dots

v_{4} = 3.32442 \dots : Δ v_{4} = 3.66071 \dots

f (v_{3}) = 6 (- 0.33628 \dots)^{3} - 6.94845 \dots (- 0.33628 \dots)^{2} - 3.53391 \dots (- 0.33628 \dots) - 11.79731 \dots = - 11.62288 \dots

f^{^{'}} (v_{3}) = 18 (- 0.33628 \dots)^{2} - 13.89691 \dots (- 0.33628 \dots) - 3.53391 \dots = 3.17502 \dots

v_{4} = 3.32442 \dots

Δ v_{4} = ∣ 3.32442 \dots - (- 0.33628 \dots) ∣ = 3.66071 \dots

Δ v_{4} = 3.66071 \dots

v_{5} = 2.51941 \dots : Δ v_{5} = 0.80501 \dots

f (v_{4}) = 6 \cdot 3.32442 \dots^{3} - 6.94845 \dots \cdot 3.32442 \dots^{2} - 3.53391 \dots \cdot 3.32442 \dots - 11.79731 \dots = 120.10728 \dots

f^{^{'}} (v_{4}) = 18 \cdot 3.32442 \dots^{2} - 13.89691 \dots \cdot 3.32442 \dots - 3.53391 \dots = 149.19962 \dots

v_{5} = 2.51941 \dots

Δ v_{5} = ∣ 2.51941 \dots - 3.32442 \dots ∣ = 0.80501 \dots

Δ v_{5} = 0.80501 \dots

v_{6} = 2.10802 \dots : Δ v_{6} = 0.41139 \dots

f (v_{5}) = 6 \cdot 2.51941 \dots^{3} - 6.94845 \dots \cdot 2.51941 \dots^{2} - 3.53391 \dots \cdot 2.51941 \dots - 11.79731 \dots = 31.14567 \dots

f^{^{'}} (v_{5}) = 18 \cdot 2.51941 \dots^{2} - 13.89691 \dots \cdot 2.51941 \dots - 3.53391 \dots = 75.70833 \dots

v_{6} = 2.10802 \dots

Δ v_{6} = ∣ 2.10802 \dots - 2.51941 \dots ∣ = 0.41139 \dots

Δ v_{6} = 0.41139 \dots

v_{7} = 1.97907 \dots : Δ v_{7} = 0.12895 \dots

f (v_{6}) = 6 \cdot 2.10802 \dots^{3} - 6.94845 \dots \cdot 2.10802 \dots^{2} - 3.53391 \dots \cdot 2.10802 \dots - 11.79731 \dots = 6.08132 \dots

f^{^{'}} (v_{6}) = 18 \cdot 2.10802 \dots^{2} - 13.89691 \dots \cdot 2.10802 \dots - 3.53391 \dots = 47.15903 \dots

v_{7} = 1.97907 \dots

Δ v_{7} = ∣ 1.97907 \dots - 2.10802 \dots ∣ = 0.12895 \dots

Δ v_{7} = 0.12895 \dots

v_{8} = 1.96633 \dots : Δ v_{8} = 0.01273 \dots

f (v_{7}) = 6 \cdot 1.97907 \dots^{3} - 6.94845 \dots \cdot 1.97907 \dots^{2} - 3.53391 \dots \cdot 1.97907 \dots - 11.79731 \dots = 0.50256 \dots

f^{^{'}} (v_{7}) = 18 \cdot 1.97907 \dots^{2} - 13.89691 \dots \cdot 1.97907 \dots - 3.53391 \dots = 39.46425 \dots

v_{8} = 1.96633 \dots

Δ v_{8} = ∣ 1.96633 \dots - 1.97907 \dots ∣ = 0.01273 \dots

Δ v_{8} = 0.01273 \dots

v_{9} = 1.96621 \dots : Δ v_{9} = 0.00011 \dots

f (v_{8}) = 6 \cdot 1.96633 \dots^{3} - 6.94845 \dots \cdot 1.96633 \dots^{2} - 3.53391 \dots \cdot 1.96633 \dots - 11.79731 \dots = 0.00463 \dots

f^{^{'}} (v_{8}) = 18 \cdot 1.96633 \dots^{2} - 13.89691 \dots \cdot 1.96633 \dots - 3.53391 \dots = 38.73684 \dots

v_{9} = 1.96621 \dots

Δ v_{9} = ∣ 1.96621 \dots - 1.96633 \dots ∣ = 0.00011 \dots

Δ v_{9} = 0.00011 \dots

v_{10} = 1.96621 \dots : Δ v_{10} = 1.05283 E - 8

f (v_{9}) = 6 \cdot 1.96621 \dots^{3} - 6.94845 \dots \cdot 1.96621 \dots^{2} - 3.53391 \dots \cdot 1.96621 \dots - 11.79731 \dots = 4.0776 E - 7

f^{^{'}} (v_{9}) = 18 \cdot 1.96621 \dots^{2} - 13.89691 \dots \cdot 1.96621 \dots - 3.53391 \dots = 38.73003 \dots

v_{10} = 1.96621 \dots

Δ v_{10} = ∣ 1.96621 \dots - 1.96621 \dots ∣ = 1.05283 E - 8

Δ v_{10} = 1.05283 E - 8

v \approx 1.96621 \dots

Apply long division:

\frac{6 v ^{3} - 6.94845 \dots v ^{2} - 3.53391 \dots v - 11.79731 \dots}{v - 1.96621 \dots} = 6 v^{2} + 4.84885 \dots v + 6

6 v^{2} + 4.84885 \dots v + 6 \approx 0

Find one solution for

6 v^{2} + 4.84885 \dots v + 6 = 0

using Newton-Raphson:

No Solution for

v \in R

6 v^{2} + 4.84885 \dots v + 6 = 0

Newton-Raphson Approximation Definition

f (v) = 6 v^{2} + 4.84885 \dots v + 6

Find

f^{^{'}} (v) : 12 v + 4.84885 \dots

\frac{d}{d v} (6 v^{2} + 4.84885 \dots v + 6)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (6 v^{2}) + \frac{d}{d v} (4.84885 \dots v) + \frac{d}{d v} (6)

\frac{d}{d v} (6 v^{2}) = 12 v

\frac{d}{d v} (6 v^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d}{d v} (v^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 6 \cdot 2 v^{2 - 1}

Simplify

= 12 v

\frac{d}{d v} (4.84885 \dots v) = 4.84885 \dots

\frac{d}{d v} (4.84885 \dots v)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4.84885 \dots \frac{d v}{d v}

Apply the common derivative:

\frac{d v}{d v} = 1

= 4.84885 \dots \cdot 1

Simplify

= 4.84885 \dots

\frac{d}{d v} (6) = 0

\frac{d}{d v} (6)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 12 v + 4.84885 \dots + 0

Simplify

= 12 v + 4.84885 \dots

Let

v_{0} = - 1

Compute

v_{n + 1}

until

Δ v_{n + 1} < 0.000001

v_{1} = 0 : Δ v_{1} = 1

f (v_{0}) = 6 (- 1)^{2} + 4.84885 \dots (- 1) + 6 = 7.15114 \dots

f^{^{'}} (v_{0}) = 12 (- 1) + 4.84885 \dots = - 7.15114 \dots

v_{1} = 0

Δ v_{1} = ∣ 0 - (- 1) ∣ = 1

Δ v_{1} = 1

v_{2} = - 1.23740 \dots : Δ v_{2} = 1.23740 \dots

f (v_{1}) = 6 \cdot 0^{2} + 4.84885 \dots \cdot 0 + 6 = 6

f^{^{'}} (v_{1}) = 12 \cdot 0 + 4.84885 \dots = 4.84885 \dots

v_{2} = - 1.23740 \dots

Δ v_{2} = ∣ - 1.23740 \dots - 0 ∣ = 1.23740 \dots

Δ v_{2} = 1.23740 \dots

v_{3} = - 0.31870 \dots : Δ v_{3} = 0.91870 \dots

f (v_{2}) = 6 (- 1.23740 \dots)^{2} + 4.84885 \dots (- 1.23740 \dots) + 6 = 9.18702 \dots

f^{^{'}} (v_{2}) = 12 (- 1.23740 \dots) + 4.84885 \dots = - 10

v_{3} = - 0.31870 \dots

Δ v_{3} = ∣ - 0.31870 \dots - (- 1.23740 \dots) ∣ = 0.91870 \dots

Δ v_{3} = 0.91870 \dots

v_{4} = - 5.26202 \dots : Δ v_{4} = 4.94332 \dots

f (v_{3}) = 6 (- 0.31870 \dots)^{2} + 4.84885 \dots (- 0.31870 \dots) + 6 = 5.06408 \dots

f^{^{'}} (v_{3}) = 12 (- 0.31870 \dots) + 4.84885 \dots = 1.02442 \dots

v_{4} = - 5.26202 \dots

Δ v_{4} = ∣ - 5.26202 \dots - (- 0.31870 \dots) ∣ = 4.94332 \dots

Δ v_{4} = 4.94332 \dots

v_{5} = - 2.74693 \dots : Δ v_{5} = 2.51509 \dots

f (v_{4}) = 6 (- 5.26202 \dots)^{2} + 4.84885 \dots (- 5.26202 \dots) + 6 = 146.61875 \dots

f^{^{'}} (v_{4}) = 12 (- 5.26202 \dots) + 4.84885 \dots = - 58.29546 \dots

v_{5} = - 2.74693 \dots

Δ v_{5} = ∣ - 2.74693 \dots - (- 5.26202 \dots) ∣ = 2.51509 \dots

Δ v_{5} = 2.51509 \dots

v_{6} = - 1.39693 \dots : Δ v_{6} = 1.34999 \dots

f (v_{5}) = 6 (- 2.74693 \dots)^{2} + 4.84885 \dots (- 2.74693 \dots) + 6 = 37.95427 \dots

f^{^{'}} (v_{5}) = 12 (- 2.74693 \dots) + 4.84885 \dots = - 28.11430 \dots

v_{6} = - 1.39693 \dots

Δ v_{6} = ∣ - 1.39693 \dots - (- 2.74693 \dots) ∣ = 1.34999 \dots

Δ v_{6} = 1.34999 \dots

v_{7} = - 0.47912 \dots : Δ v_{7} = 0.91780 \dots

f (v_{6}) = 6 (- 1.39693 \dots)^{2} + 4.84885 \dots (- 1.39693 \dots) + 6 = 10.93498 \dots

f^{^{'}} (v_{6}) = 12 (- 1.39693 \dots) + 4.84885 \dots = - 11.91431 \dots

v_{7} = - 0.47912 \dots

Δ v_{7} = ∣ - 0.47912 \dots - (- 1.39693 \dots) ∣ = 0.91780 \dots

Δ v_{7} = 0.91780 \dots

v_{8} = 5.13225 \dots : Δ v_{8} = 5.61138 \dots

f (v_{7}) = 6 (- 0.47912 \dots)^{2} + 4.84885 \dots (- 0.47912 \dots) + 6 = 5.05415 \dots

f^{^{'}} (v_{7}) = 12 (- 0.47912 \dots) + 4.84885 \dots = - 0.90069 \dots

v_{8} = 5.13225 \dots

Δ v_{8} = ∣ 5.13225 \dots - (- 0.47912 \dots) ∣ = 5.61138 \dots

Δ v_{8} = 5.61138 \dots

v_{9} = 2.28852 \dots : Δ v_{9} = 2.84373 \dots

f (v_{8}) = 6 \cdot 5.13225 \dots^{2} + 4.84885 \dots \cdot 5.13225 \dots + 6 = 188.92585 \dots

f^{^{'}} (v_{8}) = 12 \cdot 5.13225 \dots + 4.84885 \dots = 66.43592 \dots

v_{9} = 2.28852 \dots

Δ v_{9} = ∣ 2.28852 \dots - 5.13225 \dots ∣ = 2.84373 \dots

Δ v_{9} = 2.84373 \dots

v_{10} = 0.78685 \dots : Δ v_{10} = 1.50167 \dots

f (v_{9}) = 6 \cdot 2.28852 \dots^{2} + 4.84885 \dots \cdot 2.28852 \dots + 6 = 48.52081 \dots

f^{^{'}} (v_{9}) = 12 \cdot 2.28852 \dots + 4.84885 \dots = 32.31115 \dots

v_{10} = 0.78685 \dots

Δ v_{10} = ∣ 0.78685 \dots - 2.28852 \dots ∣ = 1.50167 \dots

Δ v_{10} = 1.50167 \dots

v_{11} = - 0.15990 \dots : Δ v_{11} = 0.94675 \dots

f (v_{10}) = 6 \cdot 0.78685 \dots^{2} + 4.84885 \dots \cdot 0.78685 \dots + 6 = 13.53014 \dots

f^{^{'}} (v_{10}) = 12 \cdot 0.78685 \dots + 4.84885 \dots = 14.29107 \dots

v_{11} = - 0.15990 \dots

Δ v_{11} = ∣ - 0.15990 \dots - 0.78685 \dots ∣ = 0.94675 \dots

Δ v_{11} = 0.94675 \dots

v_{12} = - 1.99540 \dots : Δ v_{12} = 1.83550 \dots

f (v_{11}) = 6 (- 0.15990 \dots)^{2} + 4.84885 \dots (- 0.15990 \dots) + 6 = 5.37806 \dots

f^{^{'}} (v_{11}) = 12 (- 0.15990 \dots) + 4.84885 \dots = 2.93001 \dots

v_{12} = - 1.99540 \dots

Δ v_{12} = ∣ - 1.99540 \dots - (- 0.15990 \dots) ∣ = 1.83550 \dots

Δ v_{12} = 1.83550 \dots

Cannot find solution

The solutions are

v \approx 0.50859 \dots, v \approx 1.96621 \dots

v \approx 0.50859 \dots, v \approx 1.96621 \dots

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

0.6 v^{2} + \frac{0.6}{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 \approx 0.50859 \dots, v \approx 1.96621 \dots

v \approx 0.50859 \dots, v \approx 1.96621 \dots

Substitute back

solve for

u

Solve

Take both sides of the equation to the power of

5 : u = 0.03402 \dots

Expand

Apply radical rule:

= (u^{\frac{1}{5}})^{5}

Apply exponent rule:

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

= u^{\frac{1}{5} \cdot 5}

\frac{1}{5} \cdot 5 = 1

\frac{1}{5} \cdot 5

Multiply fractions:

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

= \frac{1 \cdot 5}{5}

Cancel the common factor:

5

= 1

= u

Expand

0.50859 \dots^{5} : 0.03402 \dots

0.50859 \dots^{5}

0.50859 \dots^{5} = 0.03402 \dots

= 0.03402 \dots

u = 0.03402 \dots

u = 0.03402 \dots

Verify Solutions:

u = 0.03402 \dots

True

Check the solutions by plugging them into

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

Plug in

u = 0.03402 \dots :

True

= 0.50859 \dots

0.50859 \dots = 0.50859 \dots

True

The solution is

u = 0.03402 \dots

Solve

Take both sides of the equation to the power of

5 : u = 29.38731 \dots

Expand

Apply radical rule:

= (u^{\frac{1}{5}})^{5}

Apply exponent rule:

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

= u^{\frac{1}{5} \cdot 5}

\frac{1}{5} \cdot 5 = 1

\frac{1}{5} \cdot 5

Multiply fractions:

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

= \frac{1 \cdot 5}{5}

Cancel the common factor:

5

= 1

= u

Expand

1.96621 \dots^{5} : 29.38731 \dots

1.96621 \dots^{5}

1.96621 \dots^{5} = 29.38731 \dots

= 29.38731 \dots

u = 29.38731 \dots

u = 29.38731 \dots

Verify Solutions:

u = 29.38731 \dots

True

Check the solutions by plugging them into

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

Plug in

u = 29.38731 \dots :

True

= 1.96621 \dots

1.96621 \dots = 1.96621 \dots

True

The solution is

u = 29.38731 \dots

u = 0.03402 \dots, u = 29.38731 \dots

Verify Solutions:

u = 0.03402 \dots

True

, u = 29.38731 \dots

True

Check the solutions by plugging them into

\frac{u ^{0.2} + u ^{- 0.2}}{2} = \frac{0.6 ( u ^{0.4} + u ^{- 0.4} )}{2}

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

Plug in

u = 0.03402 \dots :

True

\frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{2} = \frac{0.6 ( 0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4} )}{2}

\frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{2} = 1.23740 \dots

\frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{2}

0.03402 \dots^{0.2} = 0.50859 \dots

= \frac{0.50859 \dots + 0.03402 \dots ^{- 0.2}}{2}

0.03402 \dots^{- 0.2} = 1.96621 \dots

= \frac{0.50859 \dots + 1.96621 \dots}{2}

Add the numbers:

0.50859 \dots + 1.96621 \dots = 2.47480 \dots

= \frac{2.47480 \dots}{2}

Divide the numbers:

\frac{2.47480 \dots}{2} = 1.23740 \dots

= 1.23740 \dots

\frac{0.6 ( 0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4} )}{2} = 1.23740 \dots

\frac{0.6 ( 0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4} )}{2}

Divide the numbers:

\frac{0.6}{2} = 0.3

= 0.3 (0.03402 \dots^{0.4} + 0.03402 \dots^{- 0.4})

0.03402 \dots^{0.4} = 0.25866 \dots

= 0.3 (0.25866 \dots + 0.03402 \dots^{- 0.4})

0.03402 \dots^{- 0.4} = 3.86601 \dots

= 0.3 (0.25866 \dots + 3.86601 \dots)

Add the numbers:

0.25866 \dots + 3.86601 \dots = 4.12468 \dots

= 0.3 \cdot 4.12468 \dots

Multiply the numbers:

0.3 \cdot 4.12468 \dots = 1.23740 \dots

= 1.23740 \dots

1.23740 \dots = 1.23740 \dots

True

Plug in

u = 29.38731 \dots :

True

\frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{2} = \frac{0.6 ( 29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4} )}{2}

\frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{2} = 1.23740 \dots

\frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{2}

29.38731 \dots^{0.2} = 1.96621 \dots

= \frac{1.96621 \dots + 29.38731 \dots ^{- 0.2}}{2}

29.38731 \dots^{- 0.2} = 0.50859 \dots

= \frac{1.96621 \dots + 0.50859 \dots}{2}

Add the numbers:

1.96621 \dots + 0.50859 \dots = 2.47480 \dots

= \frac{2.47480 \dots}{2}

Divide the numbers:

\frac{2.47480 \dots}{2} = 1.23740 \dots

= 1.23740 \dots

\frac{0.6 ( 29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4} )}{2} = 1.23740 \dots

\frac{0.6 ( 29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4} )}{2}

Divide the numbers:

\frac{0.6}{2} = 0.3

= 0.3 (29.38731 \dots^{0.4} + 29.38731 \dots^{- 0.4})

29.38731 \dots^{0.4} = 3.86601 \dots

= 0.3 (3.86601 \dots + 29.38731 \dots^{- 0.4})

29.38731 \dots^{- 0.4} = 0.25866 \dots

= 0.3 (0.25866 \dots + 3.86601 \dots)

Add the numbers:

3.86601 \dots + 0.25866 \dots = 4.12468 \dots

= 0.3 \cdot 4.12468 \dots

Multiply the numbers:

0.3 \cdot 4.12468 \dots = 1.23740 \dots

= 1.23740 \dots

1.23740 \dots = 1.23740 \dots

True

The solutions are

u = 0.03402 \dots, u = 29.38731 \dots

u = 0.03402 \dots, u = 29.38731 \dots

Substitute back

u = e^{m},

solve for

m

Solve

e^{m} = 0.03402 \dots : m = ln (0.03402 \dots)

e^{m} = 0.03402 \dots

Apply exponent rules

e^{m} = 0.03402 \dots

f (x) = g (x)

, then

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

ln (e^{m}) = ln (0.03402 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{m}) = m

m = ln (0.03402 \dots)

m = ln (0.03402 \dots)

Solve

e^{m} = 29.38731 \dots : m = ln (29.38731 \dots)

e^{m} = 29.38731 \dots

Apply exponent rules

e^{m} = 29.38731 \dots

f (x) = g (x)

, then

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

ln (e^{m}) = ln (29.38731 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{m}) = m

m = ln (29.38731 \dots)

m = ln (29.38731 \dots)

m = ln (0.03402 \dots), m = ln (29.38731 \dots)

Verify Solutions:

m = ln (0.03402 \dots)

True

, m = ln (29.38731 \dots)

True

Check the solutions by plugging them into

\frac{\frac{e ^{0.2 m} + e ^{- 0.2 m}}{2}}{\frac{e ^{0.4 m} + e ^{- 0.4 m}}{2}} = 0.6

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

Plug in

m = ln (0.03402 \dots) :

True

\frac{\frac{e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)}}{2}}{\frac{e ^{0.4 l n (0.03402 \dots)} + e ^{- 0.4 l n (0.03402 \dots)}}{2}} = 0.6

\frac{\frac{e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)}}{2}}{\frac{e ^{0.4 l n (0.03402 \dots)} + e ^{- 0.4 l n (0.03402 \dots)}}{2}} = 0.6

\frac{\frac{e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)}}{2}}{\frac{e ^{0.4 l n (0.03402 \dots)} + e ^{- 0.4 l n (0.03402 \dots)}}{2}}

Divide fractions:

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

= \frac{( e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)} ) \cdot 2}{2 ( e ^{0.4 l n (0.03402 \dots)} + e ^{- 0.4 l n (0.03402 \dots)} )}

Cancel the common factor:

2

= \frac{e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)}}{e ^{0.4 l n (0.03402 \dots)} + e ^{- 0.4 l n (0.03402 \dots)}}

e^{0.4 l n (0.03402 \dots)} = 0.03402 \dots^{0.4}

e^{0.4 l n (0.03402 \dots)}

Apply exponent rule:

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

= (e^{l n (0.03402 \dots)})^{0.4}

Apply log rule:

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

e^{l n (0.03402 \dots)} = 0.03402 \dots

= 0.03402 \dots^{0.4}

e^{- 0.4 l n (0.03402 \dots)} = 0.03402 \dots^{- 0.4}

e^{- 0.4 l n (0.03402 \dots)}

Apply exponent rule:

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

= (e^{l n (0.03402 \dots)})^{- 0.4}

Apply log rule:

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

e^{l n (0.03402 \dots)} = 0.03402 \dots

= 0.03402 \dots^{- 0.4}

= \frac{e ^{0.2 l n (0.03402 \dots)} + e ^{- 0.2 l n (0.03402 \dots)}}{0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4}}

e^{0.2 l n (0.03402 \dots)} = 0.03402 \dots^{0.2}

e^{0.2 l n (0.03402 \dots)}

Apply exponent rule:

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

= (e^{l n (0.03402 \dots)})^{0.2}

Apply log rule:

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

e^{l n (0.03402 \dots)} = 0.03402 \dots

= 0.03402 \dots^{0.2}

e^{- 0.2 l n (0.03402 \dots)} = 0.03402 \dots^{- 0.2}

e^{- 0.2 l n (0.03402 \dots)}

Apply exponent rule:

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

= (e^{l n (0.03402 \dots)})^{- 0.2}

Apply log rule:

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

e^{l n (0.03402 \dots)} = 0.03402 \dots

= 0.03402 \dots^{- 0.2}

= \frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4}}

Simplify

\frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{0.03402 \dots ^{0.4} + 0.03402 \dots ^{- 0.4}}

Apply exponent rule:

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

0.03402 \dots^{- 0.4} = \frac{1}{0.03402 \dots ^{0.4}}

= \frac{0.03402 \dots ^{0.2} + 0.03402 \dots ^{- 0.2}}{0.03402 \dots ^{0.4} + \frac{1}{0.03402 \dots ^{0.4}}}

Apply exponent rule:

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

0.03402 \dots^{- 0.2} = \frac{1}{0.03402 \dots ^{0.2}}

= \frac{0.03402 \dots ^{0.2} + \frac{1}{0.03402 \dots ^{0.2}}}{0.03402 \dots ^{0.4} + \frac{1}{0.03402 \dots ^{0.4}}}

Join

0.03402 \dots^{0.4} + \frac{1}{0.03402 \dots ^{0.4}} : 4.12468 \dots

0.03402 \dots^{0.4} + \frac{1}{0.03402 \dots ^{0.4}}

Convert element to fraction:

0.03402 \dots^{0.4} = \frac{0.03402 \dots ^{0.4} \cdot 0.03402 \dots ^{0.4}}{0.03402 \dots ^{0.4}}

= \frac{0.03402 \dots ^{0.4} \cdot 0.03402 \dots ^{0.4}}{0.03402 \dots ^{0.4}} + \frac{1}{0.03402 \dots ^{0.4}}

Since the denominators are equal, combine the fractions:

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

= \frac{0.03402 \dots ^{0.4} \cdot 0.03402 \dots ^{0.4} + 1}{0.03402 \dots ^{0.4}}

0.03402 \dots^{0.4} \cdot 0.03402 \dots^{0.4} + 1 = 0.03402 \dots^{0.8} + 1

0.03402 \dots^{0.4} \cdot 0.03402 \dots^{0.4} + 1

0.03402 \dots^{0.4} \cdot 0.03402 \dots^{0.4} = 0.03402 \dots^{0.8}

0.03402 \dots^{0.4} \cdot 0.03402 \dots^{0.4}

Apply exponent rule:

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

0.03402 \dots^{0.4} \cdot 0.03402 \dots^{0.4} = 0.03402 \dots^{0.4 + 0.4}

= 0.03402 \dots^{0.4 + 0.4}

Add the numbers:

0.4 + 0.4 = 0.8

= 0.03402 \dots^{0.8}

= 0.03402 \dots^{0.8} + 1

= \frac{0.03402 \dots ^{0.8} + 1}{0.03402 \dots ^{0.4}}

0.03402 \dots^{0.8} = 0.06690 \dots

= \frac{0.06690 \dots + 1}{0.03402 \dots ^{0.4}}

Add the numbers:

0.06690 \dots + 1 = 1.06690 \dots

= \frac{1.06690 \dots}{0.03402 \dots ^{0.4}}

0.03402 \dots^{0.4} = 0.25866 \dots

= \frac{1.06690 \dots}{0.25866 \dots}

Divide the numbers:

\frac{1.06690 \dots}{0.25866 \dots} = 4.12468 \dots

= 4.12468 \dots

= \frac{0.03402 \dots ^{0.2} + \frac{1}{0.03402 \dots ^{0.2}}}{4.12468 \dots}

Join

0.03402 \dots^{0.2} + \frac{1}{0.03402 \dots ^{0.2}} : 2.47480 \dots

0.03402 \dots^{0.2} + \frac{1}{0.03402 \dots ^{0.2}}

Convert element to fraction:

0.03402 \dots^{0.2} = \frac{0.03402 \dots ^{0.2} \cdot 0.03402 \dots ^{0.2}}{0.03402 \dots ^{0.2}}

= \frac{0.03402 \dots ^{0.2} \cdot 0.03402 \dots ^{0.2}}{0.03402 \dots ^{0.2}} + \frac{1}{0.03402 \dots ^{0.2}}

Since the denominators are equal, combine the fractions:

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

= \frac{0.03402 \dots ^{0.2} \cdot 0.03402 \dots ^{0.2} + 1}{0.03402 \dots ^{0.2}}

0.03402 \dots^{0.2} \cdot 0.03402 \dots^{0.2} + 1 = 0.03402 \dots^{0.4} + 1

0.03402 \dots^{0.2} \cdot 0.03402 \dots^{0.2} + 1

0.03402 \dots^{0.2} \cdot 0.03402 \dots^{0.2} = 0.03402 \dots^{0.4}

0.03402 \dots^{0.2} \cdot 0.03402 \dots^{0.2}

Apply exponent rule:

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

0.03402 \dots^{0.2} \cdot 0.03402 \dots^{0.2} = 0.03402 \dots^{0.2 + 0.2}

= 0.03402 \dots^{0.2 + 0.2}

Add the numbers:

0.2 + 0.2 = 0.4

= 0.03402 \dots^{0.4}

= 0.03402 \dots^{0.4} + 1

= \frac{0.03402 \dots ^{0.4} + 1}{0.03402 \dots ^{0.2}}

0.03402 \dots^{0.4} = 0.25866 \dots

= \frac{0.25866 \dots + 1}{0.03402 \dots ^{0.2}}

Add the numbers:

0.25866 \dots + 1 = 1.25866 \dots

= \frac{1.25866 \dots}{0.03402 \dots ^{0.2}}

0.03402 \dots^{0.2} = 0.50859 \dots

= \frac{1.25866 \dots}{0.50859 \dots}

Divide the numbers:

\frac{1.25866 \dots}{0.50859 \dots} = 2.47480 \dots

= 2.47480 \dots

= \frac{2.47480 \dots}{4.12468 \dots}

Divide the numbers:

\frac{2.47480 \dots}{4.12468 \dots} = 0.6

= 0.6

= 0.6

0.6 = 0.6

True

Plug in

m = ln (29.38731 \dots) :

True

\frac{\frac{e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)}}{2}}{\frac{e ^{0.4 l n (29.38731 \dots)} + e ^{- 0.4 l n (29.38731 \dots)}}{2}} = 0.6

\frac{\frac{e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)}}{2}}{\frac{e ^{0.4 l n (29.38731 \dots)} + e ^{- 0.4 l n (29.38731 \dots)}}{2}} = 0.6

\frac{\frac{e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)}}{2}}{\frac{e ^{0.4 l n (29.38731 \dots)} + e ^{- 0.4 l n (29.38731 \dots)}}{2}}

Divide fractions:

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

= \frac{( e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)} ) \cdot 2}{2 ( e ^{0.4 l n (29.38731 \dots)} + e ^{- 0.4 l n (29.38731 \dots)} )}

Cancel the common factor:

2

= \frac{e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)}}{e ^{0.4 l n (29.38731 \dots)} + e ^{- 0.4 l n (29.38731 \dots)}}

e^{0.4 l n (29.38731 \dots)} = 29.38731 \dots^{0.4}

e^{0.4 l n (29.38731 \dots)}

Apply exponent rule:

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

= (e^{l n (29.38731 \dots)})^{0.4}

Apply log rule:

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

e^{l n (29.38731 \dots)} = 29.38731 \dots

= 29.38731 \dots^{0.4}

e^{- 0.4 l n (29.38731 \dots)} = 29.38731 \dots^{- 0.4}

e^{- 0.4 l n (29.38731 \dots)}

Apply exponent rule:

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

= (e^{l n (29.38731 \dots)})^{- 0.4}

Apply log rule:

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

e^{l n (29.38731 \dots)} = 29.38731 \dots

= 29.38731 \dots^{- 0.4}

= \frac{e ^{0.2 l n (29.38731 \dots)} + e ^{- 0.2 l n (29.38731 \dots)}}{29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4}}

e^{0.2 l n (29.38731 \dots)} = 29.38731 \dots^{0.2}

e^{0.2 l n (29.38731 \dots)}

Apply exponent rule:

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

= (e^{l n (29.38731 \dots)})^{0.2}

Apply log rule:

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

e^{l n (29.38731 \dots)} = 29.38731 \dots

= 29.38731 \dots^{0.2}

e^{- 0.2 l n (29.38731 \dots)} = 29.38731 \dots^{- 0.2}

e^{- 0.2 l n (29.38731 \dots)}

Apply exponent rule:

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

= (e^{l n (29.38731 \dots)})^{- 0.2}

Apply log rule:

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

e^{l n (29.38731 \dots)} = 29.38731 \dots

= 29.38731 \dots^{- 0.2}

= \frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4}}

Simplify

\frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{29.38731 \dots ^{0.4} + 29.38731 \dots ^{- 0.4}}

Apply exponent rule:

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

29.38731 \dots^{- 0.4} = \frac{1}{29.38731 \dots ^{0.4}}

= \frac{29.38731 \dots ^{0.2} + 29.38731 \dots ^{- 0.2}}{29.38731 \dots ^{0.4} + \frac{1}{29.38731 \dots ^{0.4}}}

Apply exponent rule:

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

29.38731 \dots^{- 0.2} = \frac{1}{29.38731 \dots ^{0.2}}

= \frac{29.38731 \dots ^{0.2} + \frac{1}{29.38731 \dots ^{0.2}}}{29.38731 \dots ^{0.4} + \frac{1}{29.38731 \dots ^{0.4}}}

Join

29.38731 \dots^{0.4} + \frac{1}{29.38731 \dots ^{0.4}} : 4.12468 \dots

29.38731 \dots^{0.4} + \frac{1}{29.38731 \dots ^{0.4}}

Convert element to fraction:

29.38731 \dots^{0.4} = \frac{29.38731 \dots ^{0.4} \cdot 29.38731 \dots ^{0.4}}{29.38731 \dots ^{0.4}}

= \frac{29.38731 \dots ^{0.4} \cdot 29.38731 \dots ^{0.4}}{29.38731 \dots ^{0.4}} + \frac{1}{29.38731 \dots ^{0.4}}

Since the denominators are equal, combine the fractions:

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

= \frac{29.38731 \dots ^{0.4} \cdot 29.38731 \dots ^{0.4} + 1}{29.38731 \dots ^{0.4}}

29.38731 \dots^{0.4} \cdot 29.38731 \dots^{0.4} + 1 = 29.38731 \dots^{0.8} + 1

29.38731 \dots^{0.4} \cdot 29.38731 \dots^{0.4} + 1

29.38731 \dots^{0.4} \cdot 29.38731 \dots^{0.4} = 29.38731 \dots^{0.8}

29.38731 \dots^{0.4} \cdot 29.38731 \dots^{0.4}

Apply exponent rule:

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

29.38731 \dots^{0.4} \cdot 29.38731 \dots^{0.4} = 29.38731 \dots^{0.4 + 0.4}

= 29.38731 \dots^{0.4 + 0.4}

Add the numbers:

0.4 + 0.4 = 0.8

= 29.38731 \dots^{0.8}

= 29.38731 \dots^{0.8} + 1

= \frac{29.38731 \dots ^{0.8} + 1}{29.38731 \dots ^{0.4}}

29.38731 \dots^{0.8} = 14.94610 \dots

= \frac{14.94610 \dots + 1}{29.38731 \dots ^{0.4}}

Add the numbers:

14.94610 \dots + 1 = 15.94610 \dots

= \frac{15.94610 \dots}{29.38731 \dots ^{0.4}}

29.38731 \dots^{0.4} = 3.86601 \dots

= \frac{15.94610 \dots}{3.86601 \dots}

Divide the numbers:

\frac{15.94610 \dots}{3.86601 \dots} = 4.12468 \dots

= 4.12468 \dots

= \frac{29.38731 \dots ^{0.2} + \frac{1}{29.38731 \dots ^{0.2}}}{4.12468 \dots}

Join

29.38731 \dots^{0.2} + \frac{1}{29.38731 \dots ^{0.2}} : 2.47480 \dots

29.38731 \dots^{0.2} + \frac{1}{29.38731 \dots ^{0.2}}

Convert element to fraction:

29.38731 \dots^{0.2} = \frac{29.38731 \dots ^{0.2} \cdot 29.38731 \dots ^{0.2}}{29.38731 \dots ^{0.2}}

= \frac{29.38731 \dots ^{0.2} \cdot 29.38731 \dots ^{0.2}}{29.38731 \dots ^{0.2}} + \frac{1}{29.38731 \dots ^{0.2}}

Since the denominators are equal, combine the fractions:

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

= \frac{29.38731 \dots ^{0.2} \cdot 29.38731 \dots ^{0.2} + 1}{29.38731 \dots ^{0.2}}

29.38731 \dots^{0.2} \cdot 29.38731 \dots^{0.2} + 1 = 29.38731 \dots^{0.4} + 1

29.38731 \dots^{0.2} \cdot 29.38731 \dots^{0.2} + 1

29.38731 \dots^{0.2} \cdot 29.38731 \dots^{0.2} = 29.38731 \dots^{0.4}

29.38731 \dots^{0.2} \cdot 29.38731 \dots^{0.2}

Apply exponent rule:

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

29.38731 \dots^{0.2} \cdot 29.38731 \dots^{0.2} = 29.38731 \dots^{0.2 + 0.2}

= 29.38731 \dots^{0.2 + 0.2}

Add the numbers:

0.2 + 0.2 = 0.4

= 29.38731 \dots^{0.4}

= 29.38731 \dots^{0.4} + 1

= \frac{29.38731 \dots ^{0.4} + 1}{29.38731 \dots ^{0.2}}

29.38731 \dots^{0.4} = 3.86601 \dots

= \frac{3.86601 \dots + 1}{29.38731 \dots ^{0.2}}

Add the numbers:

3.86601 \dots + 1 = 4.86601 \dots

= \frac{4.86601 \dots}{29.38731 \dots ^{0.2}}

29.38731 \dots^{0.2} = 1.96621 \dots

= \frac{4.86601 \dots}{1.96621 \dots}

Divide the numbers:

\frac{4.86601 \dots}{1.96621 \dots} = 2.47480 \dots

= 2.47480 \dots

= \frac{2.47480 \dots}{4.12468 \dots}

Divide the numbers:

\frac{2.47480 \dots}{4.12468 \dots} = 0.6

= 0.6

= 0.6

0.6 = 0.6

True

The solutions are

m = ln (0.03402 \dots), m = ln (29.38731 \dots)

m = ln (0.03402 \dots), m = ln (29.38731 \dots)

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Frequently Asked Questions (FAQ)

What is the general solution for (cosh(0.2m))/(cosh(0.4m))=0.6 ?
The general solution for (cosh(0.2m))/(cosh(0.4m))=0.6 is m=ln(0.03402…),m=ln(29.38731…)