Basic Math Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = y$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $z^{- 2}$ as ${(z^{- 1})}^{2}$ .

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

Step 2.2

Rewrite $y^{- 2}$ as ${(y^{- 1})}^{2}$ .

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.3

To write $\frac{1}{z}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 2.4.4

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

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

Step 2.4.5

Write each expression with a common denominator of $z y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.5.1

Multiply $\frac{1}{z}$ by $\frac{y}{y}$ .

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

Step 2.4.5.2

Multiply $\frac{1}{y}$ by $\frac{z}{z}$ .

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

Step 2.4.5.3

Reorder the factors of $z y$ .

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

Step 2.4.6

Combine the numerators over the common denominator.

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

Step 2.4.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.8

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.9

To write $\frac{1}{z}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 2.4.10

To write $- \frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

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

Step 2.4.11

Write each expression with a common denominator of $z y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.11.1

Multiply $\frac{1}{z}$ by $\frac{y}{y}$ .

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

Step 2.4.11.2

Multiply $\frac{1}{y}$ by $\frac{z}{z}$ .

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

Step 2.4.11.3

Reorder the factors of $z y$ .

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

Step 2.4.12

Combine the numerators over the common denominator.

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

Step 3

Multiply $\frac{y + z}{y z}$ by $\frac{y - z}{y z}$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Expand $(z + y) (z - y)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify terms.

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Step 7.1

Combine the opposite terms in $z \cdot z + z (- y) + y z + y (- y)$ .

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Step 7.1.1

Reorder the factors in the terms $z (- y)$ and $y z$ .

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

Step 7.1.2

Add $- y z$ and $y z$ .

$(z \cdot z + 0 + y (- y)) \frac{y^{2} z^{2}}{(y + z) (y - z)}$

Step 7.1.3

Add $z \cdot z$ and $0$ .

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

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $z$ by $z$ .

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

Step 7.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.2.3

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

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Step 7.2.3.1

Move $y$ .

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

Step 7.2.3.2

Multiply $y$ by $y$ .

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

Step 7.3

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

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

Step 8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = y$ .

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $z + y$ and $y + z$ .

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Step 9.1.1

Reorder terms.

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

Step 9.1.2

Cancel the common factor.

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

Step 9.1.3

Rewrite the expression.

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

Step 9.2

Cancel the common factor of $z - y$ and $y - z$ .

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Step 9.2.1

Factor $- 1$ out of $z$ .

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

Step 9.2.2

Factor $- 1$ out of $- y$ .

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

Step 9.2.3

Factor $- 1$ out of $- 1 (- z) - (y)$ .

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

Step 9.2.4

Reorder terms.

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

Step 9.2.5

Cancel the common factor.

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

Step 9.2.6

Divide $- 1 y^{2} z^{2}$ by $1$ .

$- 1 y^{2} z^{2}$

Step 9.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$- y^{2} z^{2}$