Basic Math Examples

$3 = 81 \cdot r^{10 - 1}$

Step 1

Rewrite the equation as $81 \cdot r^{10 - 1} = 3$ .

$81 \cdot r^{10 - 1} = 3$

Step 2

Subtract $1$ from $10$ .

$81 \cdot r^{9} = 3$

Step 3

Divide each term in $81 \cdot r^{9} = 3$ by $81$ and simplify.

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Step 3.1

Divide each term in $81 \cdot r^{9} = 3$ by $81$ .

$\frac{81 \cdot r^{9}}{81} = \frac{3}{81}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $81$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{81 \cdot r^{9}}{81} = \frac{3}{81}$

Step 3.2.1.2

Divide $r^{9}$ by $1$ .

$r^{9} = \frac{3}{81}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $3$ and $81$ .

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Step 3.3.1.1

Factor $3$ out of $3$ .

$r^{9} = \frac{3 (1)}{81}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $3$ out of $81$ .

$r^{9} = \frac{3 \cdot 1}{3 \cdot 27}$

Step 3.3.1.2.2

Cancel the common factor.

$r^{9} = \frac{3 \cdot 1}{3 \cdot 27}$

Step 3.3.1.2.3

Rewrite the expression.

$r^{9} = \frac{1}{27}$

Step 4

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

$r = \sqrt[9]{\frac{1}{27}}$

Step 5

Simplify $\sqrt[9]{\frac{1}{27}}$ .

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Step 5.1

Rewrite $1$ as $1^{3}$ .

$r = \sqrt[9]{\frac{1^{3}}{27}}$

Step 5.2

Rewrite $27$ as $3^{3}$ .

$r = \sqrt[9]{\frac{1^{3}}{3^{3}}}$

Step 5.3

Rewrite $\frac{1^{3}}{3^{3}}$ as ${(\frac{1}{3})}^{3}$ .

$r = \sqrt[9]{{(\frac{1}{3})}^{3}}$

Step 5.4

Rewrite $\sqrt[9]{{(\frac{1}{3})}^{3}}$ as $\sqrt[3]{\sqrt[3]{{(\frac{1}{3})}^{3}}}$ .

$r = \sqrt[3]{\sqrt[3]{{(\frac{1}{3})}^{3}}}$

Step 5.5

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

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

Step 5.6

Rewrite $\sqrt[3]{\frac{1}{3}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{3}}$ .

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

Step 5.7

Any root of $1$ is $1$ .

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

Step 5.8

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

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

Step 5.9

Combine and simplify the denominator.

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Step 5.9.1

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

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

Step 5.9.2

Raise $\sqrt[3]{3}$ to the power of $1$ .

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

Step 5.9.3

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

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

Step 5.9.4

Add $1$ and $2$ .

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

Step 5.9.5

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

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Step 5.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3}$ as $3^{\frac{1}{3}}$ .

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

Step 5.9.5.2

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

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

Step 5.9.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 5.9.5.4

Cancel the common factor of $3$ .

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Step 5.9.5.4.1

Cancel the common factor.

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

Step 5.9.5.4.2

Rewrite the expression.

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

Step 5.9.5.5

Evaluate the exponent.

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

Step 5.10

Simplify the numerator.

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Step 5.10.1

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

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

Step 5.10.2

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

$r = \frac{\sqrt[3]{9}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = \frac{\sqrt[3]{9}}{3}$

Decimal Form:

$r = 0.69336127 \dots$