Solve for x x^x=1/( fourth root of 2)

$x^{x} = \frac{1}{\sqrt[4]{2}}$

Step 1

Simplify $\frac{1}{\sqrt[4]{2}}$ .

Multiply $\frac{1}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$x^{x} = \frac{1}{\sqrt[4]{2}} \cdot \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{\sqrt[4]{2} {\sqrt[4]{2}}^{3}}$

Raise $\sqrt[4]{2}$ to the power of $1$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1} {\sqrt[4]{2}}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{1 + 3}}$

Add $1$ and $3$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{4}}$

Rewrite ${\sqrt[4]{2}}^{4}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2}$ as $2^{\frac{1}{4}}$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{{(2^{\frac{1}{4}})}^{4}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{2^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{2^{\frac{4}{4}}}$

Rewrite the expression.

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{2^{1}}$

Evaluate the exponent.

$x^{x} = \frac{{\sqrt[4]{2}}^{3}}{2}$

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as ${(2^{3})}^{\frac{1}{4}}$ .

$x^{x} = \frac{\sqrt[4]{2^{3}}}{2}$

Raise $2$ to the power of $3$ .

$x^{x} = \frac{\sqrt[4]{8}}{2}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.0625, 0.80669379$

Step 3