Algebra Examples

$\frac{x y + x z}{x z - z^{2}} \times \frac{x y - y z}{x z + y z} \div \frac{y^{2} + y z}{x^{2} + x y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $x$ out of $x y + x z$ .

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Step 2.1.1

Factor $x$ out of $x y$ .

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

Step 2.1.2

Factor $x$ out of $x z$ .

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

Step 2.1.3

Factor $x$ out of $x (y) + x (z)$ .

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

Step 2.2

Factor $z$ out of $x z - z^{2}$ .

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Step 2.2.1

Factor $z$ out of $x z$ .

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

Step 2.2.2

Factor $z$ out of $- z^{2}$ .

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

Step 2.2.3

Factor $z$ out of $z x + z (- z)$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Factor $y$ out of $x y - y z$ .

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Step 3.1.1

Factor $y$ out of $x y$ .

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

Step 3.1.2

Factor $y$ out of $- y z$ .

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 4

Simplify terms.

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Step 4.1

Factor $z$ out of $x z + y z$ .

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Step 4.1.1

Factor $z$ out of $x z$ .

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

Step 4.1.2

Factor $z$ out of $y z$ .

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

Step 4.1.3

Factor $z$ out of $z x + z y$ .

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

Step 4.2

Cancel the common factor of $x - z$ .

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Step 4.2.1

Factor $x - z$ out of $z (x - z)$ .

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.3

Multiply $\frac{x (y + z)}{z}$ by $\frac{y}{z (x + y)}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Add $1$ and $1$ .

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Factor $x$ out of $x y$ .

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

Step 9.3

Factor $x$ out of $x \cdot x + x (y)$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

Factor $y$ out of $y z$ .

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

Step 10.3

Factor $y$ out of $y \cdot y + y (z)$ .

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

Step 11

Combine.

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

Step 12

Multiply $x$ by $x$ by adding the exponents.

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Step 12.1

Move $x$ .

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

Step 12.2

Multiply $x$ by $x$ .

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

Step 13

Cancel the common factor of $y + z$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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

Step 14

Cancel the common factor of $y$ .

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Step 14.1

Cancel the common factor.

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

Step 14.2

Rewrite the expression.

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

Step 15

Cancel the common factor of $x + y$ .

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Step 15.1

Cancel the common factor.

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

Step 15.2

Rewrite the expression.

$\frac{x^{2}}{z^{2}}$