What is the answer to x^x=\sqrt[4]{0.5} ?

The answer to x^x=\sqrt[4]{0.5} is x= 1/16 ,x=(ln(0.5)}{4\W_{0}(\frac{ln(0.5))/4)}

Solve for x: x^x=\sqrt[4]{0.5}

The solution is x= 1/16 ,x=(ln(0.5)}{4\W_{0}(\frac{ln(0.5))/4)}

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x = \frac{1}{16}, x = \frac{ln ( 0.5 )}{4 W _{0} ( \frac{l n ( 0.5 )}{4} )}

Solution steps

Prepare

for Lambert form:

x e^{- \frac{l n ( 0.5 )}{4 x}} = 1

Rewrite the equation with

\frac{ln ( 0.5 )}{4 x} = u

and

x = \frac{ln ( 0.5 )}{4 u}

(\frac{ln ( 0.5 )}{4 u}) e^{- u} = 1

Rewrite

(\frac{ln ( 0.5 )}{4 u}) e^{- u} = 1

in Lambert form:

e^{u} u = \frac{ln ( 0.5 )}{4}

Solve

e^{u} u = \frac{ln ( 0.5 )}{4} : u = 4 ln (0.5), u = W_{0} (\frac{ln ( 0.5 )}{4})

Verify Solutions:

u = 4 ln (0.5)

True

, u = W_{0} (\frac{ln ( 0.5 )}{4})

True

The solutions are

u = 4 ln (0.5), u = W_{0} (\frac{ln ( 0.5 )}{4})

Substitute back

u = \frac{ln ( 0.5 )}{4 x},

solve for

x

Solve

\frac{ln ( 0.5 )}{4 x} = 4 ln (0.5) : x = \frac{1}{16}

Solve

\frac{ln ( 0.5 )}{4 x} = W_{0} (\frac{ln ( 0.5 )}{4}) : x = \frac{ln ( 0.5 )}{4 W _{0} ( \frac{l n ( 0.5 )}{4} )}

x = \frac{1}{16}, x = \frac{ln ( 0.5 )}{4 W _{0} ( \frac{l n ( 0.5 )}{4} )}

0 = e^{x} + x e^{x}

3^{x} = 2187

12 = 200 (1 - e^{- \frac{1}{800} t})

7 = 5 e^{0.2 x}

731093 = 525600 e^{0.03 t}

What is the answer to x^x=\sqrt[4]{0.5} ?
The answer to x^x=\sqrt[4]{0.5} is x= 1/16 ,x=(ln(0.5)}{4\W_{0}(\frac{ln(0.5))/4)}
Solve for x: x^x=\sqrt[4]{0.5}
The solution is x= 1/16 ,x=(ln(0.5)}{4\W_{0}(\frac{ln(0.5))/4)}