What is the general solution for cos^3(3θ)= 1/4 ?

The general solution for cos^3(3θ)= 1/4 is θ=(0.88929…)/3+(2pin)/3 ,θ=(2pi)/3-(0.88929…)/3+(2pin)/3

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cos^3(3θ)= 1/4

Solution

cos^{3} (3 θ) = \frac{1}{4}

Solution

θ = \frac{0.88929 \dots}{3} + \frac{2 πn}{3}, θ = \frac{2 π}{3} - \frac{0.88929 \dots}{3} + \frac{2 πn}{3}

Degrees

θ = 16.98426 \dots^{\circ} + 12 0^{\circ} n, θ = 103.01573 \dots^{\circ} + 12 0^{\circ} n

Solution steps

cos^{3} (3 θ) = \frac{1}{4}

Solve by substitution

cos^{3} (3 θ) = \frac{1}{4}

Let:

cos (3 θ) = u

u^{3} = \frac{1}{4}

u^{3} = \frac{1}{4}

For

x^{3} = f (a)

the solutions are

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

Multiply:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

Remove parentheses:

(- a) = - a

= - 1 + 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

\frac{4 ^{\frac{2}{3}}}{4 ^{\frac{2}{3}}}

Apply exponent rule:

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

= 4^{\frac{2}{3} + \frac{1}{3}} \cdot 2

4^{\frac{2}{3} + \frac{1}{3}} = 4

4^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 4^{1}

Apply rule

a^{1} = a

= 4

= 4 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 8

= \frac{4 ^{\frac{2}{3}} ( - 1 + 3 i )}{8}

= \frac{4 ^{\frac{2}{3}} ( - 1 + 3 i )}{8}

Rewrite

\frac{4 ^{\frac{2}{3}} ( - 1 + 3 i )}{8}

in standard complex form:

- \frac{4 ^{\frac{2}{3}}}{8} + \frac{4 ^{\frac{2}{3}} 3}{8} i

\frac{4 ^{\frac{2}{3}} ( - 1 + 3 i )}{8}

Expand

4^{\frac{2}{3}} (- 1 + 3 i) : - 4^{\frac{2}{3}} + 4^{\frac{2}{3}} 3 i

4^{\frac{2}{3}} (- 1 + 3 i)

Apply the distributive law:

a (b + c) = ab + a c

a = 4^{\frac{2}{3}}, b = - 1, c = 3 i

= 4^{\frac{2}{3}} (- 1) + 4^{\frac{2}{3}} 3 i

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 4^{\frac{2}{3}} + 4^{\frac{2}{3}} 3 i

Multiply:

1 \cdot 4^{\frac{2}{3}} = 4^{\frac{2}{3}}

= - 4^{\frac{2}{3}} + 4^{\frac{2}{3}} 3 i

= \frac{- 4 ^{\frac{2}{3}} + 4 ^{\frac{2}{3}} 3 i}{8}

Apply the fraction rule:

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

\frac{- 4 ^{\frac{2}{3}} + 4 ^{\frac{2}{3}} 3 i}{8} = - \frac{4 ^{\frac{2}{3}}}{8} + \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{4 ^{\frac{2}{3}}}{8} + \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{4 ^{\frac{2}{3}}}{8} + \frac{4 ^{\frac{2}{3}} 3}{8} i

Simplify

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Apply rule

Multiply

Multiply fractions:

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

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

Multiply:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

Remove parentheses:

(- a) = - a

= - 1 - 3 i

Apply the fraction rule:

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

Rationalize

Multiply by the conjugate

\frac{4 ^{\frac{2}{3}}}{4 ^{\frac{2}{3}}}

Apply exponent rule:

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

= 4^{\frac{2}{3} + \frac{1}{3}} \cdot 2

4^{\frac{2}{3} + \frac{1}{3}} = 4

4^{\frac{2}{3} + \frac{1}{3}}

Combine the fractions

\frac{2}{3} + \frac{1}{3} : 1

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 4^{1}

Apply rule

a^{1} = a

= 4

= 4 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 8

= \frac{4 ^{\frac{2}{3}} ( - 1 - 3 i )}{8}

= \frac{4 ^{\frac{2}{3}} ( - 1 - 3 i )}{8}

Rewrite

\frac{4 ^{\frac{2}{3}} ( - 1 - 3 i )}{8}

in standard complex form:

- \frac{4 ^{\frac{2}{3}}}{8} - \frac{4 ^{\frac{2}{3}} 3}{8} i

\frac{4 ^{\frac{2}{3}} ( - 1 - 3 i )}{8}

Expand

4^{\frac{2}{3}} (- 1 - 3 i) : - 4^{\frac{2}{3}} - 4^{\frac{2}{3}} 3 i

4^{\frac{2}{3}} (- 1 - 3 i)

Apply the distributive law:

a (b - c) = ab - a c

a = 4^{\frac{2}{3}}, b = - 1, c = 3 i

= 4^{\frac{2}{3}} (- 1) - 4^{\frac{2}{3}} 3 i

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 4^{\frac{2}{3}} - 4^{\frac{2}{3}} 3 i

Multiply:

1 \cdot 4^{\frac{2}{3}} = 4^{\frac{2}{3}}

= - 4^{\frac{2}{3}} - 4^{\frac{2}{3}} 3 i

= \frac{- 4 ^{\frac{2}{3}} - 4 ^{\frac{2}{3}} 3 i}{8}

Apply the fraction rule:

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

\frac{- 4 ^{\frac{2}{3}} - 4 ^{\frac{2}{3}} 3 i}{8} = - \frac{4 ^{\frac{2}{3}}}{8} - \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{4 ^{\frac{2}{3}}}{8} - \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{4 ^{\frac{2}{3}}}{8} - \frac{4 ^{\frac{2}{3}} 3}{8} i

Substitute back

u = cos (3 θ)

Apply trig inverse properties

General solutions for

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

Solve

Divide both sides by

3

Divide both sides by

3

Simplify

Solve

Divide both sides by

3

Divide both sides by

3

Simplify

cos (3 θ) = - \frac{4 ^{\frac{2}{3}}}{8} + i \frac{4 ^{\frac{2}{3}} 3}{8} :

No Solution

cos (3 θ) = - \frac{4 ^{\frac{2}{3}}}{8} + i \frac{4 ^{\frac{2}{3}} 3}{8}

Simplify

- \frac{4 ^{\frac{2}{3}}}{8} + i \frac{4 ^{\frac{2}{3}} 3}{8}

Cancel

\frac{4 ^{\frac{2}{3}}}{8}

Factor

4^{\frac{2}{3}} : 2^{\frac{4}{3}}

Factor

4 = 2^{2}

= (2^{2})^{\frac{2}{3}}

Simplify

(2^{2})^{\frac{2}{3}} : 2^{\frac{4}{3}}

(2^{2})^{\frac{2}{3}}

Apply exponent rule:

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

assuming

a \geq 0

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

2 \cdot \frac{2}{3} = \frac{4}{3}

2 \cdot \frac{2}{3}

Multiply fractions:

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

= \frac{2 \cdot 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{3}

= 2^{\frac{4}{3}}

= 2^{\frac{4}{3}}

Factor

8 : 2^{3}

Factor

8 = 2^{3}

= \frac{2 ^{\frac{4}{3}}}{2 ^{3}}

Cancel

\frac{2 ^{\frac{4}{3}}}{2 ^{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}, 2^{3} = 2^{1 + 2}

= \frac{2 ^{1 + \frac{1}{3}}}{2 ^{1 + 2}}

Apply exponent rule:

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

2^{1 + \frac{1}{3}} = 2^{1} \cdot 2^{\frac{1}{3}}, 2^{1 + 2} = 2^{1} \cdot 2^{2}

= \frac{2 ^{1} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{1}}

Cancel the common factor:

2^{1}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

2^{2} = 4

Cancel

Factor

4 : 2^{2}

Factor

4 = 2^{2}

Cancel

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

Rewrite

in standard complex form:

Apply radical rule:

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

Apply exponent rule:

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

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

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

Subtract the numbers:

2 - \frac{1}{3} = \frac{5}{3}

= \frac{1}{2 ^{\frac{5}{3}}}

2^{\frac{5}{3}} = 2 \cdot 2^{\frac{2}{3}}

2^{\frac{5}{3}}

2^{\frac{5}{3}} = 2^{1 + \frac{2}{3}}

= 2^{1 + \frac{2}{3}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{2}{3}}

Refine

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

= \frac{1}{2 \cdot 2 ^{\frac{2}{3}}}

i \frac{4 ^{\frac{2}{3}} 3}{8} = \frac{4 ^{\frac{2}{3}} 3 i}{8}

i \frac{4 ^{\frac{2}{3}} 3}{8}

Multiply fractions:

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

= \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} + \frac{4 ^{\frac{2}{3}} 3 i}{8}

Least Common Multiplier of

2 2^{\frac{2}{3}}, 8 : 8 \cdot 2^{\frac{2}{3}}

2 \cdot 2^{\frac{2}{3}}, 8

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Compute an expression comprised of factors that appear either in

2 2^{\frac{2}{3}}

8

= 8 \cdot 2^{\frac{2}{3}}

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8 \cdot 2^{\frac{2}{3}}

For

\frac{1}{2 \cdot 2 ^{\frac{2}{3}}} :

multiply the denominator and numerator by

4

\frac{1}{2 \cdot 2 ^{\frac{2}{3}}} = \frac{1 \cdot 4}{2 \cdot 2 ^{\frac{2}{3}} \cdot 4} = \frac{4}{8 \cdot 2 ^{\frac{2}{3}}}

For

\frac{4 ^{\frac{2}{3}} 3 i}{8} :

multiply the denominator and numerator by

2^{\frac{2}{3}}

\frac{4 ^{\frac{2}{3}} 3 i}{8} = \frac{4 ^{\frac{2}{3}} 3 i 2 ^{\frac{2}{3}}}{8 \cdot 2 ^{\frac{2}{3}}} = \frac{3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

= - \frac{4}{8 \cdot 2 ^{\frac{2}{3}}} + \frac{3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 + 3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

3 \cdot 2^{\frac{4}{3} + \frac{2}{3}} i = 43 i

3 \cdot 2^{\frac{4}{3} + \frac{2}{3}} i

2^{\frac{4}{3} + \frac{2}{3}} = 2^{2}

2^{\frac{4}{3} + \frac{2}{3}}

Join

\frac{4}{3} + \frac{2}{3} : 2

\frac{4}{3} + \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 2}{3}

Add the numbers:

4 + 2 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

= 2^{2} 3 i

2^{2} = 4

= 43 i

= \frac{- 4 + 4 3 i}{8 \cdot 2 ^{\frac{2}{3}}}

Factor

- 4 + 34 i : 4 (- 1 + 3 i)

- 4 + 3 \cdot 4 i

Rewrite as

= - 4 \cdot 1 + 43 i

Factor out common term

4

= 4 (- 1 + 3 i)

= \frac{4 ( - 1 + 3 i )}{8 \cdot 2 ^{\frac{2}{3}}}

Cancel the common factor:

4

= \frac{- 1 + 3 i}{2 \cdot 2 ^{\frac{2}{3}}}

Apply the fraction rule:

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

\frac{- 1 + 3 i}{2 \cdot 2 ^{\frac{2}{3}}} = - \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} + \frac{3 i}{2 \cdot 2 ^{\frac{2}{3}}}

= - \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} + \frac{3 i}{2 \cdot 2 ^{\frac{2}{3}}}

\frac{3}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

Apply exponent rule:

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

= 2^{1 + \frac{2}{3} + \frac{1}{3}}

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

1 + \frac{2}{3} + \frac{1}{3}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

\frac{1}{1} = \frac{1 \cdot 3}{1 \cdot 3} = \frac{3}{3}

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

- \frac{1}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

Apply exponent rule:

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

= 2^{1 + \frac{2}{3} + \frac{1}{3}}

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

1 + \frac{2}{3} + \frac{1}{3}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

\frac{1}{1} = \frac{1 \cdot 3}{1 \cdot 3} = \frac{3}{3}

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

No Solution

cos (3 θ) = - \frac{4 ^{\frac{2}{3}}}{8} - i \frac{4 ^{\frac{2}{3}} 3}{8} :

No Solution

cos (3 θ) = - \frac{4 ^{\frac{2}{3}}}{8} - i \frac{4 ^{\frac{2}{3}} 3}{8}

Simplify

- \frac{4 ^{\frac{2}{3}}}{8} - i \frac{4 ^{\frac{2}{3}} 3}{8}

Cancel

\frac{4 ^{\frac{2}{3}}}{8}

Factor

4^{\frac{2}{3}} : 2^{\frac{4}{3}}

Factor

4 = 2^{2}

= (2^{2})^{\frac{2}{3}}

Simplify

(2^{2})^{\frac{2}{3}} : 2^{\frac{4}{3}}

(2^{2})^{\frac{2}{3}}

Apply exponent rule:

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

assuming

a \geq 0

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

2 \cdot \frac{2}{3} = \frac{4}{3}

2 \cdot \frac{2}{3}

Multiply fractions:

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

= \frac{2 \cdot 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4}{3}

= 2^{\frac{4}{3}}

= 2^{\frac{4}{3}}

Factor

8 : 2^{3}

Factor

8 = 2^{3}

= \frac{2 ^{\frac{4}{3}}}{2 ^{3}}

Cancel

\frac{2 ^{\frac{4}{3}}}{2 ^{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}, 2^{3} = 2^{1 + 2}

= \frac{2 ^{1 + \frac{1}{3}}}{2 ^{1 + 2}}

Apply exponent rule:

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

2^{1 + \frac{1}{3}} = 2^{1} \cdot 2^{\frac{1}{3}}, 2^{1 + 2} = 2^{1} \cdot 2^{2}

= \frac{2 ^{1} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{1}}

Cancel the common factor:

2^{1}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

2^{2} = 4

Cancel

Factor

4 : 2^{2}

Factor

4 = 2^{2}

Cancel

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

= \frac{2 ^{0 + \frac{1}{3}}}{2 ^{0 + 2}}

Apply exponent rule:

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

2^{0 + \frac{1}{3}} = 2^{0} \cdot 2^{\frac{1}{3}}, 2^{0 + 2} = 2^{0} \cdot 2^{2}

= \frac{2 ^{0} \cdot 2 ^{\frac{1}{3}}}{2 ^{2} \cdot 2 ^{0}}

Cancel the common factor:

2^{0}

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

Rewrite

in standard complex form:

Apply radical rule:

= \frac{2 ^{\frac{1}{3}}}{2 ^{2}}

Apply exponent rule:

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

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

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

Subtract the numbers:

2 - \frac{1}{3} = \frac{5}{3}

= \frac{1}{2 ^{\frac{5}{3}}}

2^{\frac{5}{3}} = 2 \cdot 2^{\frac{2}{3}}

2^{\frac{5}{3}}

2^{\frac{5}{3}} = 2^{1 + \frac{2}{3}}

= 2^{1 + \frac{2}{3}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{2}{3}}

Refine

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

= \frac{1}{2 \cdot 2 ^{\frac{2}{3}}}

i \frac{4 ^{\frac{2}{3}} 3}{8} = \frac{4 ^{\frac{2}{3}} 3 i}{8}

i \frac{4 ^{\frac{2}{3}} 3}{8}

Multiply fractions:

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

= \frac{4 ^{\frac{2}{3}} 3 i}{8}

= - \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} - \frac{4 ^{\frac{2}{3}} 3 i}{8}

Least Common Multiplier of

2 2^{\frac{2}{3}}, 8 : 8 \cdot 2^{\frac{2}{3}}

2 \cdot 2^{\frac{2}{3}}, 8

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Compute an expression comprised of factors that appear either in

2 2^{\frac{2}{3}}

8

= 8 \cdot 2^{\frac{2}{3}}

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8 \cdot 2^{\frac{2}{3}}

For

\frac{1}{2 \cdot 2 ^{\frac{2}{3}}} :

multiply the denominator and numerator by

4

\frac{1}{2 \cdot 2 ^{\frac{2}{3}}} = \frac{1 \cdot 4}{2 \cdot 2 ^{\frac{2}{3}} \cdot 4} = \frac{4}{8 \cdot 2 ^{\frac{2}{3}}}

For

\frac{4 ^{\frac{2}{3}} 3 i}{8} :

multiply the denominator and numerator by

2^{\frac{2}{3}}

\frac{4 ^{\frac{2}{3}} 3 i}{8} = \frac{4 ^{\frac{2}{3}} 3 i 2 ^{\frac{2}{3}}}{8 \cdot 2 ^{\frac{2}{3}}} = \frac{3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

= - \frac{4}{8 \cdot 2 ^{\frac{2}{3}}} - \frac{3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 - 3 \cdot 2 ^{\frac{4}{3} + \frac{2}{3}} i}{8 \cdot 2 ^{\frac{2}{3}}}

3 \cdot 2^{\frac{4}{3} + \frac{2}{3}} i = 43 i

3 \cdot 2^{\frac{4}{3} + \frac{2}{3}} i

2^{\frac{4}{3} + \frac{2}{3}} = 2^{2}

2^{\frac{4}{3} + \frac{2}{3}}

Join

\frac{4}{3} + \frac{2}{3} : 2

\frac{4}{3} + \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{4 + 2}{3}

Add the numbers:

4 + 2 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

= 2^{2} 3 i

2^{2} = 4

= 43 i

= \frac{- 4 - 4 3 i}{8 \cdot 2 ^{\frac{2}{3}}}

Factor

- 4 - 34 i : - 4 (1 + 3 i)

- 4 - 3 \cdot 4 i

Rewrite as

= - 4 \cdot 1 - 43 i

Factor out common term

4

= - 4 (1 + 3 i)

= - \frac{4 ( 1 + 3 i )}{8 \cdot 2 ^{\frac{2}{3}}}

Cancel the common factor:

4

= - \frac{1 + 3 i}{2 \cdot 2 ^{\frac{2}{3}}}

Apply the fraction rule:

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

\frac{1 + 3 i}{2 \cdot 2 ^{\frac{2}{3}}} = - (\frac{1}{2 \cdot 2 ^{\frac{2}{3}}}) - (\frac{3 i}{2 \cdot 2 ^{\frac{2}{3}}})

= - (\frac{1}{2 \cdot 2 ^{\frac{2}{3}}}) - (\frac{3 i}{2 \cdot 2 ^{\frac{2}{3}}})

Remove parentheses:

(a) = a

= - \frac{1}{2 \cdot 2 ^{\frac{2}{3}}} - \frac{3 i}{2 \cdot 2 ^{\frac{2}{3}}}

- \frac{3}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

Apply exponent rule:

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

= 2^{1 + \frac{2}{3} + \frac{1}{3}}

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

1 + \frac{2}{3} + \frac{1}{3}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

\frac{1}{1} = \frac{1 \cdot 3}{1 \cdot 3} = \frac{3}{3}

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

- \frac{1}{2 \cdot 2 ^{\frac{2}{3}}}

Multiply by the conjugate

Apply exponent rule:

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

= 2^{1 + \frac{2}{3} + \frac{1}{3}}

Join

1 + \frac{2}{3} + \frac{1}{3} : 2

1 + \frac{2}{3} + \frac{1}{3}

Convert element to fraction:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

Least Common Multiplier of

1, 3, 3 : 3

1, 3, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 3, 3

= 3

Multiply the numbers:

3 = 3

= 3

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3

For

\frac{1}{1} :

multiply the denominator and numerator by

3

\frac{1}{1} = \frac{1 \cdot 3}{1 \cdot 3} = \frac{3}{3}

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 + 2 + 1}{3}

Add the numbers:

3 + 2 + 1 = 6

= \frac{6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

No Solution

Combine all the solutions

Show solutions in decimal form

θ = \frac{0.88929 \dots}{3} + \frac{2 πn}{3}, θ = \frac{2 π}{3} - \frac{0.88929 \dots}{3} + \frac{2 πn}{3}

Popular Examples

3sin(2x)-3/2 sqrt(3)=0

3 sin (2 x) - \frac{3}{2} 3 = 0

-12sin(3x)=0

- 12 sin (3 x) = 0

sin^2(θ)-1/4 =0

sin^{2} (θ) - \frac{1}{4} = 0

1=sech(x)

1 = sech (x)

sin^2(x)+3sin(x)=0

sin^{2} (x) + 3 sin (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for cos^3(3θ)= 1/4 ?
The general solution for cos^3(3θ)= 1/4 is θ=(0.88929…)/3+(2pin)/3 ,θ=(2pi)/3-(0.88929…)/3+(2pin)/3