Basic Math Examples

$\frac{1}{27} = 27 (r^{6})$

Step 1

Rewrite the equation as $27 r^{6} = \frac{1}{27}$ .

$27 r^{6} = \frac{1}{27}$

Step 2

Divide each term in $27 r^{6} = \frac{1}{27}$ by $27$ and simplify.

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Step 2.1

Divide each term in $27 r^{6} = \frac{1}{27}$ by $27$ .

$\frac{27 r^{6}}{27} = \frac{\frac{1}{27}}{27}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $27$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{27 r^{6}}{27} = \frac{\frac{1}{27}}{27}$

Step 2.2.1.2

Divide $r^{6}$ by $1$ .

$r^{6} = \frac{\frac{1}{27}}{27}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$r^{6} = \frac{1}{27} \cdot \frac{1}{27}$

Step 2.3.2

Multiply $\frac{1}{27} \cdot \frac{1}{27}$ .

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Step 2.3.2.1

Multiply $\frac{1}{27}$ by $\frac{1}{27}$ .

$r^{6} = \frac{1}{27 \cdot 27}$

Step 2.3.2.2

Multiply $27$ by $27$ .

$r^{6} = \frac{1}{729}$

Step 3

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

$r = \pm \sqrt[6]{\frac{1}{729}}$

Step 4

Simplify $\pm \sqrt[6]{\frac{1}{729}}$ .

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Step 4.1

Rewrite $\sqrt[6]{\frac{1}{729}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{729}}$ .

$r = \pm \frac{\sqrt[6]{1}}{\sqrt[6]{729}}$

Step 4.2

Any root of $1$ is $1$ .

$r = \pm \frac{1}{\sqrt[6]{729}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $729$ as $3^{6}$ .

$r = \pm \frac{1}{\sqrt[6]{3^{6}}}$

Step 4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$r = \pm \frac{1}{3}$

Step 5

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

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Step 5.1

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

$r = \frac{1}{3}$

Step 5.2

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

$r = - \frac{1}{3}$

Step 5.3

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

$r = \frac{1}{3}, - \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{1}{3}, - \frac{1}{3}$

Decimal Form:

$r = 0.\overline{3}, - 0.\overline{3}$