Algebra Examples

$\frac{\sqrt[4]{m} + \sqrt[4]{n}}{\sqrt[4]{m} - \sqrt[4]{n}} = \frac{7}{3}$

Step 1

Multiply both sides by $\sqrt[4]{m} - \sqrt[4]{n}$ .

$\frac{\sqrt[4]{m} + \sqrt[4]{n}}{\sqrt[4]{m} - \sqrt[4]{n}} (\sqrt[4]{m} - \sqrt[4]{n}) = \frac{7}{3} (\sqrt[4]{m} - \sqrt[4]{n})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $\sqrt[4]{m} - \sqrt[4]{n}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{\sqrt[4]{m} + \sqrt[4]{n}}{\sqrt[4]{m} - \sqrt[4]{n}} (\sqrt[4]{m} - \sqrt[4]{n}) = \frac{7}{3} (\sqrt[4]{m} - \sqrt[4]{n})$

Step 2.1.1.2

Rewrite the expression.

$\sqrt[4]{m} + \sqrt[4]{n} = \frac{7}{3} (\sqrt[4]{m} - \sqrt[4]{n})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{7}{3} (\sqrt[4]{m} - \sqrt[4]{n})$ .

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Step 2.2.1.1

Apply the distributive property.

$\sqrt[4]{m} + \sqrt[4]{n} = \frac{7}{3} \sqrt[4]{m} + \frac{7}{3} (- \sqrt[4]{n})$

Step 2.2.1.2

Combine $\frac{7}{3}$ and $\sqrt[4]{m}$ .

$\sqrt[4]{m} + \sqrt[4]{n} = \frac{7 \sqrt[4]{m}}{3} + \frac{7}{3} (- \sqrt[4]{n})$

Step 2.2.1.3

Combine $\frac{7}{3}$ and $\sqrt[4]{n}$ .

$\sqrt[4]{m} + \sqrt[4]{n} = \frac{7 \sqrt[4]{m}}{3} - \frac{7 \sqrt[4]{n}}{3}$

Step 3

Solve for $m$ .

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Step 3.1

Solve for $\sqrt[4]{m}$ .

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Step 3.1.1

Move all terms containing $\sqrt[4]{m}$ to the left side of the equation.

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Step 3.1.1.1

Subtract $\frac{7 \sqrt[4]{m}}{3}$ from both sides of the equation.

$\sqrt[4]{m} + \sqrt[4]{n} - \frac{7 \sqrt[4]{m}}{3} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.2

To write $\sqrt[4]{m}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt[4]{m} \cdot \frac{3}{3} - \frac{7 \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.3

Combine $\sqrt[4]{m}$ and $\frac{3}{3}$ .

$\frac{\sqrt[4]{m} \cdot 3}{3} - \frac{7 \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.4

Combine the numerators over the common denominator.

$\frac{\sqrt[4]{m} \cdot 3 - 7 \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.5

Subtract $7 \sqrt[4]{m}$ from $\sqrt[4]{m} \cdot 3$ .

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Step 3.1.1.5.1

Reorder $\sqrt[4]{m}$ and $3$ .

$\frac{3 \cdot \sqrt[4]{m} - 7 \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.5.2

Subtract $7 \sqrt[4]{m}$ from $3 \cdot \sqrt[4]{m}$ .

$\frac{- 4 \cdot \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.1.6

Move the negative in front of the fraction.

$- \frac{4 \sqrt[4]{m}}{3} + \sqrt[4]{n} = - \frac{7 \sqrt[4]{n}}{3}$

Step 3.1.2

Subtract $\sqrt[4]{n}$ from both sides of the equation.

$- \frac{4 \sqrt[4]{m}}{3} = - \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n}$

Step 3.1.3

Multiply both sides of the equation by $- \frac{3}{4}$ .

$- \frac{3}{4} (- \frac{4 \sqrt[4]{m}}{3}) = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4

Simplify both sides of the equation.

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Step 3.1.4.1

Simplify the left side.

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Step 3.1.4.1.1

Simplify $- \frac{3}{4} (- \frac{4 \sqrt[4]{m}}{3})$ .

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Step 3.1.4.1.1.1

Cancel the common factor of $3$ .

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Step 3.1.4.1.1.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (- \frac{4 \sqrt[4]{m}}{3}) = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.1.2

Move the leading negative in $- \frac{4 \sqrt[4]{m}}{3}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 4 \sqrt[4]{m}}{3} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 4 \sqrt[4]{m}}{3} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 4 \sqrt[4]{m}}{3} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 \sqrt[4]{m}) = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.2

Cancel the common factor of $4$ .

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Step 3.1.4.1.1.2.1

Factor $4$ out of $- 4 \sqrt[4]{m}$ .

$\frac{- 1}{4} (4 (- \sqrt[4]{m})) = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- \sqrt[4]{m})) = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.2.3

Rewrite the expression.

$- - \sqrt[4]{m} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.3

Multiply.

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Step 3.1.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt[4]{m} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.1.1.3.2

Multiply $\sqrt[4]{m}$ by $1$ .

$\sqrt[4]{m} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$

Step 3.1.4.2

Simplify the right side.

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Step 3.1.4.2.1

Simplify $- \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n})$ .

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Step 3.1.4.2.1.1

To write $- \sqrt[4]{n}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt[4]{m} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} - \sqrt[4]{n} \cdot \frac{3}{3})$

Step 3.1.4.2.1.2

Combine $- \sqrt[4]{n}$ and $\frac{3}{3}$ .

$\sqrt[4]{m} = - \frac{3}{4} (- \frac{7 \sqrt[4]{n}}{3} + \frac{- \sqrt[4]{n} \cdot 3}{3})$

Step 3.1.4.2.1.3

Combine the numerators over the common denominator.

$\sqrt[4]{m} = - \frac{3}{4} \cdot \frac{- 7 \sqrt[4]{n} - \sqrt[4]{n} \cdot 3}{3}$

Step 3.1.4.2.1.4

Cancel the common factor of $3$ .

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Step 3.1.4.2.1.4.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\sqrt[4]{m} = \frac{- 3}{4} \cdot \frac{- 7 \sqrt[4]{n} - \sqrt[4]{n} \cdot 3}{3}$

Step 3.1.4.2.1.4.2

Factor $3$ out of $- 3$ .

$\sqrt[4]{m} = \frac{3 (- 1)}{4} \cdot \frac{- 7 \sqrt[4]{n} - \sqrt[4]{n} \cdot 3}{3}$

Step 3.1.4.2.1.4.3

Cancel the common factor.

$\sqrt[4]{m} = \frac{3 \cdot - 1}{4} \cdot \frac{- 7 \sqrt[4]{n} - \sqrt[4]{n} \cdot 3}{3}$

Step 3.1.4.2.1.4.4

Rewrite the expression.

$\sqrt[4]{m} = \frac{- 1}{4} (- 7 \sqrt[4]{n} - \sqrt[4]{n} \cdot 3)$

Step 3.1.4.2.1.5

Simplify the expression.

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Step 3.1.4.2.1.5.1

Multiply $3$ by $- 1$ .

$\sqrt[4]{m} = \frac{- 1}{4} (- 7 \sqrt[4]{n} - 3 \sqrt[4]{n})$

Step 3.1.4.2.1.5.2

Move the negative in front of the fraction.

$\sqrt[4]{m} = - \frac{1}{4} (- 7 \sqrt[4]{n} - 3 \sqrt[4]{n})$

Step 3.1.4.2.1.6

Subtract $3 \sqrt[4]{n}$ from $- 7 \sqrt[4]{n}$ .

$\sqrt[4]{m} = - \frac{1}{4} (- 10 \sqrt[4]{n})$

Step 3.1.4.2.1.7

Cancel the common factor of $2$ .

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Step 3.1.4.2.1.7.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\sqrt[4]{m} = \frac{- 1}{4} (- 10 \sqrt[4]{n})$

Step 3.1.4.2.1.7.2

Factor $2$ out of $4$ .

$\sqrt[4]{m} = \frac{- 1}{2 (2)} (- 10 \sqrt[4]{n})$

Step 3.1.4.2.1.7.3

Factor $2$ out of $- 10 \sqrt[4]{n}$ .

$\sqrt[4]{m} = \frac{- 1}{2 (2)} (2 (- 5 \sqrt[4]{n}))$

Step 3.1.4.2.1.7.4

Cancel the common factor.

$\sqrt[4]{m} = \frac{- 1}{2 \cdot 2} (2 (- 5 \sqrt[4]{n}))$

Step 3.1.4.2.1.7.5

Rewrite the expression.

$\sqrt[4]{m} = \frac{- 1}{2} (- 5 \sqrt[4]{n})$

Step 3.1.4.2.1.8

Combine $- 5$ and $\frac{- 1}{2}$ .

$\sqrt[4]{m} = \frac{- 5 \cdot - 1}{2} \sqrt[4]{n}$

Step 3.1.4.2.1.9

Multiply $- 5$ by $- 1$ .

$\sqrt[4]{m} = \frac{5}{2} \sqrt[4]{n}$

Step 3.1.4.2.1.10

Combine $\frac{5}{2}$ and $\sqrt[4]{n}$ .

$\sqrt[4]{m} = \frac{5 \sqrt[4]{n}}{2}$

Step 3.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{m}}^{4} = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

${(m^{\frac{1}{4}})}^{4} = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${(m^{\frac{1}{4}})}^{4}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${(m^{\frac{1}{4}})}^{4}$ .

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Step 3.3.2.1.1.1

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

$m^{\frac{1}{4} \cdot 4} = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3.2.1.1.2

Cancel the common factor of $4$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

$m^{\frac{1}{4} \cdot 4} = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

$m^{1} = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3.2.1.2

Simplify.

$m = {(\frac{5 \sqrt[4]{n}}{2})}^{4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify ${(\frac{5 \sqrt[4]{n}}{2})}^{4}$ .

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Step 3.3.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.3.1.1.1

Apply the product rule to $\frac{5 \sqrt[4]{n}}{2}$ .

$m = \frac{{(5 \sqrt[4]{n})}^{4}}{2^{4}}$

Step 3.3.3.1.1.2

Apply the product rule to $5 \sqrt[4]{n}$ .

$m = \frac{5^{4} {\sqrt[4]{n}}^{4}}{2^{4}}$

Step 3.3.3.1.2

Simplify the numerator.

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Step 3.3.3.1.2.1

Raise $5$ to the power of $4$ .

$m = \frac{625 {\sqrt[4]{n}}^{4}}{2^{4}}$

Step 3.3.3.1.2.2

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

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Step 3.3.3.1.2.2.1

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

$m = \frac{625 {(n^{\frac{1}{4}})}^{4}}{2^{4}}$

Step 3.3.3.1.2.2.2

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

$m = \frac{625 n^{\frac{1}{4} \cdot 4}}{2^{4}}$

Step 3.3.3.1.2.2.3

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

$m = \frac{625 n^{\frac{4}{4}}}{2^{4}}$

Step 3.3.3.1.2.2.4

Cancel the common factor of $4$ .

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Step 3.3.3.1.2.2.4.1

Cancel the common factor.

$m = \frac{625 n^{\frac{4}{4}}}{2^{4}}$

Step 3.3.3.1.2.2.4.2

Rewrite the expression.

$m = \frac{625 n^{1}}{2^{4}}$

Step 3.3.3.1.2.2.5

Simplify.

$m = \frac{625 n}{2^{4}}$

Step 3.3.3.1.3

Raise $2$ to the power of $4$ .

$m = \frac{625 n}{16}$