What is the answer to x^x=2^{1/6} ?

The answer to x^x=2^{1/6} is x=(ln(2)}{6\W_{0}(\frac{ln(2))/6)}

Solve for x: x^x=2^{1/6}

The solution is x=(ln(2)}{6\W_{0}(\frac{ln(2))/6)}

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x^{x} = 2^{\frac{1}{6}}

x = \frac{ln ( 2 )}{6 W _{0} ( \frac{l n ( 2 )}{6} )}

Decimal Notation

x = 1.10971 \dots

Solution steps

x^{x} = 2^{\frac{1}{6}}

Prepare

x^{x} = 2^{\frac{1}{6}}

for Lambert form:

x e^{- \frac{l n ( 2 )}{6 x}} = 1

Rewrite the equation with

\frac{ln ( 2 )}{6 x} = u

and

x = \frac{ln ( 2 )}{6 u}

(\frac{ln ( 2 )}{6 u}) e^{- u} = 1

Rewrite

(\frac{ln ( 2 )}{6 u}) e^{- u} = 1

in Lambert form:

e^{u} u = \frac{ln ( 2 )}{6}

Solve

e^{u} u = \frac{ln ( 2 )}{6} : u = W_{0} (\frac{ln ( 2 )}{6})

Verify Solutions:

u = W_{0} (\frac{ln ( 2 )}{6})

True

The solution is

u = W_{0} (\frac{ln ( 2 )}{6})

Substitute back

u = \frac{ln ( 2 )}{6 x},

solve for

x

Solve

\frac{ln ( 2 )}{6 x} = W_{0} (\frac{ln ( 2 )}{6}) : x = \frac{ln ( 2 )}{6 W _{0} ( \frac{l n ( 2 )}{6} )}

x = \frac{ln ( 2 )}{6 W _{0} ( \frac{l n ( 2 )}{6} )}

73 = 1 5^{x}

3 e^{5 x} - 2 = 10

3 \cdot 6^{2 x} = 1 2^{x}

\frac{a x}{e ^{b x}} = 0

2^{n} = 1000 \cdot 60 \cdot 60

What is the answer to x^x=2^{1/6} ?
The answer to x^x=2^{1/6} is x=(ln(2)}{6\W_{0}(\frac{ln(2))/6)}
Solve for x: x^x=2^{1/6}
The solution is x=(ln(2)}{6\W_{0}(\frac{ln(2))/6)}