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 $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$c y^{4}, 64 y^{4}$

Step 3.1.2

Since $c y^{4}, 64 y^{4}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 64$ then find LCM for the variable part $c^{1}, y^{4}, y^{4}$ .

Step 3.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 3.1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.1.5

The prime factors for $64$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 3.1.5.1

$64$ has factors of $2$ and $32$ .

$2 \cdot 32$

Step 3.1.5.2

$32$ has factors of $2$ and $16$ .

$2 \cdot 2 \cdot 16$

Step 3.1.5.3

$16$ has factors of $2$ and $8$ .

$2 \cdot 2 \cdot 2 \cdot 8$

Step 3.1.5.4

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 4$

Step 3.1.5.5

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Step 3.1.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 3.1.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

Step 3.1.6.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2$

Step 3.1.6.3

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2$

Step 3.1.6.4

Multiply $16$ by $2$ .

$32 \cdot 2$

Step 3.1.6.5

Multiply $32$ by $2$ .

$64$

Step 3.1.7

The factor for $c^{1}$ is $c$ itself.

$c^{1} = c$

$c$ occurs $1$ time.

Step 3.1.8

The factors for $y^{4}$ are $y \cdot y \cdot y \cdot y$ , which is $y$ multiplied by each other $4$ times.

$y^{4} = y \cdot y \cdot y \cdot y$

$y$ occurs $4$ times.

Step 3.1.9

The LCM of $c^{1}, y^{4}, y^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$c \cdot y \cdot y \cdot y \cdot y$

Step 3.1.10

Simplify $c \cdot y \cdot y \cdot y \cdot y$ .

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Step 3.1.10.1

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

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Step 3.1.10.1.1

Move $y$ .

$c \cdot (y \cdot y) \cdot y \cdot y$

Step 3.1.10.1.2

Multiply $y$ by $y$ .

$c \cdot y^{2} \cdot y \cdot y$

Step 3.1.10.2

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 3.1.10.2.1

Move $y$ .

$c \cdot (y \cdot y^{2}) \cdot y$

Step 3.1.10.2.2

Multiply $y$ by $y^{2}$ .

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Step 3.1.10.2.2.1

Raise $y$ to the power of $1$ .

$c \cdot (y^{1} y^{2}) \cdot y$

Step 3.1.10.2.2.2

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

$c \cdot y^{1 + 2} \cdot y$

Step 3.1.10.2.3

Add $1$ and $2$ .

$c \cdot y^{3} \cdot y$

Step 3.1.10.3

Multiply $y^{3}$ by $y$ by adding the exponents.

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Step 3.1.10.3.1

Move $y$ .

$c \cdot (y \cdot y^{3})$

Step 3.1.10.3.2

Multiply $y$ by $y^{3}$ .

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Step 3.1.10.3.2.1

Raise $y$ to the power of $1$ .

$c \cdot (y^{1} y^{3})$

Step 3.1.10.3.2.2

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

$c \cdot y^{1 + 3}$

Step 3.1.10.3.3

Add $1$ and $3$ .

$c \cdot y^{4}$

$c y^{4}$

Step 3.1.11

The LCM for $c y^{4}, 64 y^{4}$ is the numeric part $64$ multiplied by the variable part.

$64 c y^{4}$

Step 3.2

Multiply each term in $\frac{x^{3}}{c y^{4}} = \frac{x^{3}}{64 y^{4}}$ by $64 c y^{4}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x^{3}}{c y^{4}} = \frac{x^{3}}{64 y^{4}}$ by $64 c y^{4}$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.2.2

Combine $64$ and $\frac{x^{3}}{c y^{4}}$ .

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

Step 3.2.2.3

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

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Step 3.2.2.3.1

Cancel the common factor.

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

Step 3.2.2.3.2

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.3.2

Cancel the common factor of $64$ .

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Step 3.2.3.2.1

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

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

Step 3.2.3.2.2

Cancel the common factor.

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

Step 3.2.3.2.3

Rewrite the expression.

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

Step 3.2.3.3

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

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Step 3.2.3.3.1

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

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

Step 3.2.3.3.2

Cancel the common factor.

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

Step 3.2.3.3.3

Rewrite the expression.

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

Step 3.3

Solve the equation.

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 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

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

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

Step 3.3.4

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

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Step 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 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $64 - c$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

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

Step 3.3.4.2.1.2

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

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

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Divide $0$ by $64 - c$ .

$x^{3} = 0$

Step 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 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

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

$x = 0$

Step 4

The variable $y$ got canceled.

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$