Finite Math Examples

$a \cdot 4 = 3 \cdot r^{5 - 1}$

Step 1

Rewrite the equation as $3 \cdot r^{5 - 1} = a \cdot 4$ .

$3 \cdot r^{5 - 1} = a \cdot 4$

Step 2

Subtract $1$ from $5$ .

$3 \cdot r^{4} = a \cdot 4$

Step 3

Divide each term in $3 \cdot r^{4} = a \cdot 4$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 \cdot r^{4} = a \cdot 4$ by $3$ .

$\frac{3 \cdot r^{4}}{3} = \frac{a \cdot 4}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 \cdot r^{4}}{3} = \frac{a \cdot 4}{3}$

Step 3.2.1.2

Divide $r^{4}$ by $1$ .

$r^{4} = \frac{a \cdot 4}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move $4$ to the left of $a$ .

$r^{4} = \frac{4 a}{3}$

Step 4

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

$r = \pm \sqrt[4]{\frac{4 a}{3}}$

Step 5

Simplify $\pm \sqrt[4]{\frac{4 a}{3}}$ .

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Step 5.1

Rewrite $\sqrt[4]{\frac{4 a}{3}}$ as $\frac{\sqrt[4]{4 a}}{\sqrt[4]{3}}$ .

$r = \pm \frac{\sqrt[4]{4 a}}{\sqrt[4]{3}}$

Step 5.2

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

$r = \pm \frac{\sqrt[4]{4 a}}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

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

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 5.3.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 5.3.3

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

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 5.3.4

Add $1$ and $3$ .

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 5.3.5

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

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Step 5.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 5.3.5.2

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

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 5.3.5.3

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

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.3.5.4

Cancel the common factor of $4$ .

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Step 5.3.5.4.1

Cancel the common factor.

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.3.5.4.2

Rewrite the expression.

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{3^{1}}$

Step 5.3.5.5

Evaluate the exponent.

$r = \pm \frac{\sqrt[4]{4 a} {\sqrt[4]{3}}^{3}}{3}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

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

$r = \pm \frac{\sqrt[4]{4 a} \sqrt[4]{3^{3}}}{3}$

Step 5.4.2

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

$r = \pm \frac{\sqrt[4]{4 a} \sqrt[4]{27}}{3}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt[4]{4 a \cdot 27}}{3}$

Step 5.5.2

Multiply $27$ by $4$ .

$r = \pm \frac{\sqrt[4]{108 a}}{3}$

Step 6

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

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Step 6.1

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

$r = \frac{\sqrt[4]{108 a}}{3}$

Step 6.2

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

$r = - \frac{\sqrt[4]{108 a}}{3}$

Step 6.3

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

$r = \frac{\sqrt[4]{108 a}}{3}$

$r = - \frac{\sqrt[4]{108 a}}{3}$

$r = \frac{\sqrt[4]{108 a}}{3}$

$r = - \frac{\sqrt[4]{108 a}}{3}$