Algebra Examples

$x^{x} = \frac{1}{\sqrt{2}}$

Step 1

Simplify $\frac{1}{\sqrt{2}}$ .

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Step 1.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x^{x} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x^{x} = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.2

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

$x^{x} = \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.3

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

$x^{x} = \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.4

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

$x^{x} = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5

Add $1$ and $1$ .

$x^{x} = \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.6.1

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

$x^{x} = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.6.2

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

$x^{x} = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$x^{x} = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$x^{x} = \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.6.4.2

Rewrite the expression.

$x^{x} = \frac{\sqrt{2}}{2^{1}}$

Step 1.2.6.5

Evaluate the exponent.

$x^{x} = \frac{\sqrt{2}}{2}$

Step 2

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

$x = \frac{1}{4}, \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}, \frac{1}{2}$

Decimal Form:

$x = 0.25, 0.5$

Step 4