Calculus Examples

$x^{2} = \frac{1}{5}$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{5}}$

Step 2

Simplify $\pm \sqrt{\frac{1}{5}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{5}}$

Step 2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{5}}$

Step 2.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.4.4

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

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.4.6.2

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.4.6.3

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{5}}{5^{1}}$

Step 2.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{5}}{5}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{5}}{5}$

Step 3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{5}}{5}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}$

Decimal Form:

$x = 0.44721359 \dots, - 0.44721359 \dots$