Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Add $1$ and $3$ .

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

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

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

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

Step 1.2.5.3

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

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

Step 1.2.5.4

Cancel the common factor of $4$ .

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Step 1.2.5.4.1

Cancel the common factor.

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

Step 1.2.5.4.2

Rewrite the expression.

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

Step 1.2.5.5

Evaluate the exponent.

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

Step 1.3

Simplify the numerator.

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Step 1.3.1

Rewrite ${\sqrt[4]{2}}^{3}$ as $\sqrt[4]{2^{3}}$ .

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

Step 1.3.2

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