Algebra Examples

$\sqrt[3]{\frac{x^{3}}{c y^{4}}} = \frac{x}{4 y (\sqrt[3]{y})}$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{\frac{x^{3}}{c y^{4}}}}^{3} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(\frac{x^{3}}{c y^{4}})}^{\frac{1}{3}})}^{3} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(\frac{x^{3}}{c y^{4}})}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(\frac{x^{3}}{c y^{4}})}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(\frac{x^{3}}{c y^{4}})}^{\frac{1}{3} \cdot 3} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(\frac{x^{3}}{c y^{4}})}^{\frac{1}{3} \cdot 3} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(\frac{x^{3}}{c y^{4}})}^{1} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2.2.1.2

Simplify.

$\frac{x^{3}}{c y^{4}} = {(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\frac{x}{4 y (\sqrt[3]{y})})}^{3}$ .

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Step 2.3.1.1

Multiply $\frac{x}{4 y (\sqrt[3]{y})}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x}{4 y (\sqrt[3]{y})} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}})}^{3}$

Step 2.3.1.2

Combine and simplify the denominator.

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Step 2.3.1.2.1

Multiply $\frac{x}{4 y (\sqrt[3]{y})}$ by $\frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}}$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y (\sqrt[3]{y}) {\sqrt[3]{y}}^{2}})}^{3}$

Step 2.3.1.2.2

Move $\sqrt[3]{y}$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y (\sqrt[3]{y} {\sqrt[3]{y}}^{2})})}^{3}$

Step 2.3.1.2.3

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y ({\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2})})}^{3}$

Step 2.3.1.2.4

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y {\sqrt[3]{y}}^{1 + 2}})}^{3}$

Step 2.3.1.2.5

Add $1$ and $2$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y {\sqrt[3]{y}}^{3}})}^{3}$

Step 2.3.1.2.6

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

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Step 2.3.1.2.6.1

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y {(y^{\frac{1}{3}})}^{3}})}^{3}$

Step 2.3.1.2.6.2

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y \cdot y^{\frac{1}{3} \cdot 3}})}^{3}$

Step 2.3.1.2.6.3

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y \cdot y^{\frac{3}{3}}})}^{3}$

Step 2.3.1.2.6.4

Cancel the common factor of $3$ .

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Step 2.3.1.2.6.4.1

Cancel the common factor.

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y \cdot y^{\frac{3}{3}}})}^{3}$

Step 2.3.1.2.6.4.2

Rewrite the expression.

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y \cdot y^{1}})}^{3}$

Step 2.3.1.2.6.5

Simplify.

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y \cdot y})}^{3}$

Step 2.3.1.3

Multiply $y$ by $y$ by adding the exponents.

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Step 2.3.1.3.1

Move $y$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 (y \cdot y)})}^{3}$

Step 2.3.1.3.2

Multiply $y$ by $y$ .

$\frac{x^{3}}{c y^{4}} = {(\frac{x {\sqrt[3]{y}}^{2}}{4 y^{2}})}^{3}$

Step 2.3.1.4

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

$\frac{x^{3}}{c y^{4}} = {(\frac{x \sqrt[3]{y^{2}}}{4 y^{2}})}^{3}$

Step 2.3.1.5

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

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Step 2.3.1.5.1

Apply the product rule to $\frac{x \sqrt[3]{y^{2}}}{4 y^{2}}$ .

$\frac{x^{3}}{c y^{4}} = \frac{{(x \sqrt[3]{y^{2}})}^{3}}{{(4 y^{2})}^{3}}$

Step 2.3.1.5.2

Apply the product rule to $x \sqrt[3]{y^{2}}$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} {\sqrt[3]{y^{2}}}^{3}}{{(4 y^{2})}^{3}}$

Step 2.3.1.5.3

Apply the product rule to $4 y^{2}$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} {\sqrt[3]{y^{2}}}^{3}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6

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

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Step 2.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{2}}$ as $y^{\frac{2}{3}}$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} {(y^{\frac{2}{3}})}^{3}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.2

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

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{2}{3} \cdot 3}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.3

Combine $\frac{2}{3}$ and $3$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{2 \cdot 3}{3}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.4

Multiply $2$ by $3$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{6}{3}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.5

Cancel the common factor of $6$ and $3$ .

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Step 2.3.1.6.5.1

Factor $3$ out of $6$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{3 \cdot 2}{3}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.5.2

Cancel the common factors.

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Step 2.3.1.6.5.2.1

Factor $3$ out of $3$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{3 \cdot 2}{3 (1)}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.5.2.2

Cancel the common factor.

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{3 \cdot 2}{3 \cdot 1}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.5.2.3

Rewrite the expression.

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{\frac{2}{1}}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.6.5.2.4

Divide $2$ by $1$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{2}}{4^{3} {(y^{2})}^{3}}$

Step 2.3.1.7

Simplify the denominator.

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Step 2.3.1.7.1

Raise $4$ to the power of $3$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{2}}{64 {(y^{2})}^{3}}$

Step 2.3.1.7.2

Multiply the exponents in ${(y^{2})}^{3}$ .

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Step 2.3.1.7.2.1

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

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{2}}{64 y^{2 \cdot 3}}$

Step 2.3.1.7.2.2

Multiply $2$ by $3$ .

$\frac{x^{3}}{c y^{4}} = \frac{x^{3} y^{2}}{64 y^{6}}$

Step 2.3.1.8

Cancel the common factor of $y^{2}$ and $y^{6}$ .

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Step 2.3.1.8.1

Factor $y^{2}$ out of $x^{3} y^{2}$ .

$\frac{x^{3}}{c y^{4}} = \frac{y^{2} x^{3}}{64 y^{6}}$

Step 2.3.1.8.2

Cancel the common factors.

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Step 2.3.1.8.2.1

Factor $y^{2}$ out of $64 y^{6}$ .

$\frac{x^{3}}{c y^{4}} = \frac{y^{2} x^{3}}{y^{2} (64 y^{4})}$

Step 2.3.1.8.2.2

Cancel the common factor.

$\frac{x^{3}}{c y^{4}} = \frac{y^{2} x^{3}}{y^{2} (64 y^{4})}$

Step 2.3.1.8.2.3

Rewrite the expression.

$\frac{x^{3}}{c y^{4}} = \frac{x^{3}}{64 y^{4}}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $c y^{4}$ .

$\frac{x^{3}}{c y^{4}} (c y^{4}) = \frac{x^{3}}{64 y^{4}} (c y^{4})$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $c y^{4}$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{x^{3}}{c y^{4}} (c y^{4}) = \frac{x^{3}}{64 y^{4}} (c y^{4})$

Step 3.2.1.1.2

Rewrite the expression.

$x^{3} = \frac{x^{3}}{64 y^{4}} (c y^{4})$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{x^{3}}{64 y^{4}} (c y^{4})$ .

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Step 3.2.2.1.1

Cancel the common factor of $y^{4}$ .

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Step 3.2.2.1.1.1

Factor $y^{4}$ out of $64 y^{4}$ .

$x^{3} = \frac{x^{3}}{y^{4} \cdot 64} (c y^{4})$

Step 3.2.2.1.1.2

Factor $y^{4}$ out of $c y^{4}$ .

$x^{3} = \frac{x^{3}}{y^{4} \cdot 64} (y^{4} c)$

Step 3.2.2.1.1.3

Cancel the common factor.

$x^{3} = \frac{x^{3}}{y^{4} \cdot 64} (y^{4} c)$

Step 3.2.2.1.1.4

Rewrite the expression.

$x^{3} = \frac{x^{3}}{64} c$

Step 3.2.2.1.2

Combine $\frac{x^{3}}{64}$ and $c$ .

$x^{3} = \frac{x^{3} c}{64}$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Multiply both sides by $64$ .

$x^{3} \cdot 64 = \frac{x^{3} c}{64} \cdot 64$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Move $64$ to the left of $x^{3}$ .

$64 x^{3} = \frac{x^{3} c}{64} \cdot 64$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Cancel the common factor of $64$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$64 x^{3} = \frac{x^{3} c}{64} \cdot 64$

Step 3.3.2.2.1.2

Rewrite the expression.

$64 x^{3} = x^{3} c$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $x^{3} c$ from both sides of the equation.

$64 x^{3} - x^{3} c = 0$

Step 3.3.3.2

Factor $x^{3}$ out of $64 x^{3} - x^{3} c$ .

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Step 3.3.3.2.1

Factor $x^{3}$ out of $64 x^{3}$ .

$x^{3} \cdot 64 - x^{3} c = 0$

Step 3.3.3.2.2

Factor $x^{3}$ out of $- x^{3} c$ .

$x^{3} \cdot 64 + x^{3} (- 1 c) = 0$

Step 3.3.3.2.3

Factor $x^{3}$ out of $x^{3} \cdot 64 + x^{3} (- 1 c)$ .

$x^{3} (64 - 1 c) = 0$

Step 3.3.3.3

Rewrite $- 1 c$ as $- c$ .

$x^{3} (64 - c) = 0$

Step 3.3.3.4

Divide each term in $x^{3} (64 - c) = 0$ by $64 - c$ and simplify.

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Step 3.3.3.4.1

Divide each term in $x^{3} (64 - c) = 0$ by $64 - c$ .

$\frac{x^{3} (64 - c)}{64 - c} = \frac{0}{64 - c}$

Step 3.3.3.4.2

Simplify the left side.

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Step 3.3.3.4.2.1

Cancel the common factor of $64 - c$ .

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Step 3.3.3.4.2.1.1

Cancel the common factor.

$\frac{x^{3} (64 - c)}{64 - c} = \frac{0}{64 - c}$

Step 3.3.3.4.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{64 - c}$

Step 3.3.3.4.3

Simplify the right side.

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Step 3.3.3.4.3.1

Divide $0$ by $64 - c$ .

$x^{3} = 0$

Step 3.3.3.5

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

$x = \sqrt[3]{0}$

Step 3.3.3.6

Simplify $\sqrt[3]{0}$ .

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Step 3.3.3.6.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.3.3.6.2

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

$x = 0$