Basic Math Examples

$\frac{\frac{1}{z} - \frac{1}{y}}{\frac{1}{z^{3}} - \frac{1}{y^{3}}}$

Step 1

Multiply the numerator and denominator of the fraction by $y^{3} z^{3}$ .

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Step 1.1

Multiply $\frac{\frac{1}{z} - \frac{1}{y}}{\frac{1}{z^{3}} - \frac{1}{y^{3}}}$ by $\frac{y^{3} z^{3}}{y^{3} z^{3}}$ .

$\frac{y^{3} z^{3}}{y^{3} z^{3}} \cdot \frac{\frac{1}{z} - \frac{1}{y}}{\frac{1}{z^{3}} - \frac{1}{y^{3}}}$

Step 1.2

Combine.

$\frac{y^{3} z^{3} (\frac{1}{z} - \frac{1}{y})}{y^{3} z^{3} (\frac{1}{z^{3}} - \frac{1}{y^{3}})}$

Step 2

Apply the distributive property.

$\frac{y^{3} z^{3} \frac{1}{z} + y^{3} z^{3} (- \frac{1}{y})}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $z$ .

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Step 3.1.1

Factor $z$ out of $y^{3} z^{3}$ .

$\frac{z (y^{3} z^{2}) \frac{1}{z} + y^{3} z^{3} (- \frac{1}{y})}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.1.2

Cancel the common factor.

$\frac{z (y^{3} z^{2}) \frac{1}{z} + y^{3} z^{3} (- \frac{1}{y})}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.1.3

Rewrite the expression.

$\frac{y^{3} z^{2} + y^{3} z^{3} (- \frac{1}{y})}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

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

$\frac{y^{3} z^{2} + y^{3} z^{3} \frac{- 1}{y}}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.2.2

Factor $y$ out of $y^{3} z^{3}$ .

$\frac{y^{3} z^{2} + y (y^{2} z^{3}) \frac{- 1}{y}}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.2.3

Cancel the common factor.

$\frac{y^{3} z^{2} + y (y^{2} z^{3}) \frac{- 1}{y}}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.2.4

Rewrite the expression.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} z^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.3

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

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Step 3.3.1

Factor $z^{3}$ out of $y^{3} z^{3}$ .

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{z^{3} y^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.3.2

Cancel the common factor.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{z^{3} y^{3} \frac{1}{z^{3}} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.3.3

Rewrite the expression.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} + y^{3} z^{3} (- \frac{1}{y^{3}})}$

Step 3.4

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

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Step 3.4.1

Move the leading negative in $- \frac{1}{y^{3}}$ into the numerator.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} + y^{3} z^{3} \frac{- 1}{y^{3}}}$

Step 3.4.2

Factor $y^{3}$ out of $y^{3} z^{3}$ .

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} + y^{3} (z^{3}) \frac{- 1}{y^{3}}}$

Step 3.4.3

Cancel the common factor.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} + y^{3} z^{3} \frac{- 1}{y^{3}}}$

Step 3.4.4

Rewrite the expression.

$\frac{y^{3} z^{2} + y^{2} z^{3} \cdot - 1}{y^{3} + z^{3} \cdot - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $y^{2} z^{2}$ out of $y^{3} z^{2} + y^{2} z^{3} \cdot - 1$ .

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Step 4.1.1

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

$\frac{y^{2} z^{2} (y) + y^{2} z^{3} \cdot - 1}{y^{3} + z^{3} \cdot - 1}$

Step 4.1.2

Factor $y^{2} z^{2}$ out of $y^{2} z^{3} \cdot - 1$ .

$\frac{y^{2} z^{2} (y) + y^{2} z^{2} (z \cdot - 1)}{y^{3} + z^{3} \cdot - 1}$

Step 4.1.3

Factor $y^{2} z^{2}$ out of $y^{2} z^{2} (y) + y^{2} z^{2} (z \cdot - 1)$ .

$\frac{y^{2} z^{2} (y + z \cdot - 1)}{y^{3} + z^{3} \cdot - 1}$

Step 4.2

Move $- 1$ to the left of $z$ .

$\frac{y^{2} z^{2} (y - 1 \cdot z)}{y^{3} + z^{3} \cdot - 1}$

Step 4.3

Rewrite $- 1 z$ as $- z$ .

$\frac{y^{2} z^{2} (y - z)}{y^{3} + z^{3} \cdot - 1}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $z^{3} \cdot - 1$ as ${(z \cdot - 1)}^{3}$ .

$\frac{y^{2} z^{2} (y - z)}{y^{3} + {(z \cdot - 1)}^{3}}$

Step 5.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = y$ and $b = z \cdot - 1$ .

$\frac{y^{2} z^{2} (y - z)}{(y + z \cdot - 1) (y^{2} - y (z \cdot - 1) + {(z \cdot - 1)}^{2})}$

Step 5.3

Simplify.

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Step 5.3.1

Move $- 1$ to the left of $z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - 1 \cdot z) (y^{2} - y (z \cdot - 1) + {(z \cdot - 1)}^{2})}$

Step 5.3.2

Rewrite $- 1 z$ as $- z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} - y (z \cdot - 1) + {(z \cdot - 1)}^{2})}$

Step 5.3.3

Move $- 1$ to the left of $z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} - y (- 1 \cdot z) + {(z \cdot - 1)}^{2})}$

Step 5.3.4

Rewrite using the commutative property of multiplication.

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} - 1 \cdot - 1 y z + {(z \cdot - 1)}^{2})}$

Step 5.3.5

Multiply $- 1$ by $- 1$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + 1 y z + {(z \cdot - 1)}^{2})}$

Step 5.3.6

Multiply $y$ by $1$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + {(z \cdot - 1)}^{2})}$

Step 5.3.7

Move $- 1$ to the left of $z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + {(- 1 \cdot z)}^{2})}$

Step 5.3.8

Rewrite $- 1 z$ as $- z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + {(- z)}^{2})}$

Step 5.3.9

Apply the product rule to $- z$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + {(- 1)}^{2} z^{2})}$

Step 5.3.10

Raise $- 1$ to the power of $2$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + 1 z^{2})}$

Step 5.3.11

Multiply $z^{2}$ by $1$ .

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + z^{2})}$

Step 6

Cancel the common factor of $y - z$ .

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Step 6.1

Cancel the common factor.

$\frac{y^{2} z^{2} (y - z)}{(y - z) (y^{2} + y z + z^{2})}$

Step 6.2

Rewrite the expression.

$\frac{y^{2} z^{2}}{y^{2} + y z + z^{2}}$