Basic Math 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 $c$ .

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Step 3.1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 3.2

Solve the equation for $c$ .

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Step 3.2.1

Rewrite the equation as $c y^{4} x^{3} = x^{3} \cdot (64 y^{4})$ .

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

Step 3.2.2

Divide each term in $c y^{4} x^{3} = x^{3} \cdot (64 y^{4})$ by $y^{4} x^{3}$ and simplify.

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Step 3.2.2.1

Divide each term in $c y^{4} x^{3} = x^{3} \cdot (64 y^{4})$ by $y^{4} x^{3}$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

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

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Rewrite the expression.

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

Step 3.2.2.2.2

Cancel the common factor of $x^{3}$ .

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Step 3.2.2.2.2.1

Cancel the common factor.

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

Step 3.2.2.2.2.2

Divide $c$ by $1$ .

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Cancel the common factor of $x^{3}$ .

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Step 3.2.2.3.1.1

Cancel the common factor.

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

Step 3.2.2.3.1.2

Rewrite the expression.

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

Step 3.2.2.3.2

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

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Step 3.2.2.3.2.1

Cancel the common factor.

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

Step 3.2.2.3.2.2

Divide $64$ by $1$ .

$c = 64$