Algebra Examples

$\frac{15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y}{5 x y}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y$ and $g (x) = 5 x y$ .

$\frac{5 x y \frac{d}{d x} [15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y] - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 2

By the Sum Rule, the derivative of $15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y$ with respect to $x$ is $\frac{d}{d x} [15 x^{4} y^{2}] + \frac{d}{d x} [- 30 x^{2} y^{3}] + \frac{d}{d x} [45 x y]$ .

$\frac{5 x y (\frac{d}{d x} [15 x^{4} y^{2}] + \frac{d}{d x} [- 30 x^{2} y^{3}] + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [15 x^{4} y^{2}]$ .

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Step 3.1

Since $15 y^{2}$ is constant with respect to $x$ , the derivative of $15 x^{4} y^{2}$ with respect to $x$ is $15 y^{2} \frac{d}{d x} [x^{4}]$ .

$\frac{5 x y (15 y^{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 30 x^{2} y^{3}] + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{5 x y (15 y^{2} (4 x^{3}) + \frac{d}{d x} [- 30 x^{2} y^{3}] + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 3.3

Multiply $4$ by $15$ .

$\frac{5 x y (60 y^{2} x^{3} + \frac{d}{d x} [- 30 x^{2} y^{3}] + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 4

Evaluate $\frac{d}{d x} [- 30 x^{2} y^{3}]$ .

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Step 4.1

Since $- 30 y^{3}$ is constant with respect to $x$ , the derivative of $- 30 x^{2} y^{3}$ with respect to $x$ is $- 30 y^{3} \frac{d}{d x} [x^{2}]$ .

$\frac{5 x y (60 y^{2} x^{3} - 30 y^{3} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{5 x y (60 y^{2} x^{3} - 30 y^{3} (2 x) + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 4.3

Multiply $2$ by $- 30$ .

$\frac{5 x y (60 y^{2} x^{3} - 60 y^{3} x + \frac{d}{d x} [45 x y]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 5

Evaluate $\frac{d}{d x} [45 x y]$ .

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Step 5.1

Since $45 y$ is constant with respect to $x$ , the derivative of $45 x y$ with respect to $x$ is $45 y \frac{d}{d x} [x]$ .

$\frac{5 x y (60 y^{2} x^{3} - 60 y^{3} x + 45 y \frac{d}{d x} [x]) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{5 x y (60 y^{2} x^{3} - 60 y^{3} x + 45 y \cdot 1) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 5.3

Multiply $45$ by $1$ .

$\frac{5 x y (60 y^{2} x^{3} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 6

Differentiate.

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Step 6.1

Reorder terms.

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) \frac{d}{d x} [5 x y]}{{(5 x y)}^{2}}$

Step 6.2

Since $5 y$ is constant with respect to $x$ , the derivative of $5 x y$ with respect to $x$ is $5 y \frac{d}{d x} [x]$ .

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y \frac{d}{d x} [x])}{{(5 x y)}^{2}}$

Step 6.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y \cdot 1)}{{(5 x y)}^{2}}$

Step 6.4

Multiply $5$ by $1$ .

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y)}{{(5 x y)}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the product rule to $5 x y$ .

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y)}{{(5 x)}^{2} y^{2}}$

Step 7.2

Apply the product rule to $5 x$ .

$\frac{5 x y (60 x^{3} y^{2} - 60 y^{3} x + 45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.3

Apply the distributive property.

$\frac{5 x y (60 x^{3} y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2} - 30 x^{2} y^{3} + 45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.4

Apply the distributive property.

$\frac{5 x y (60 x^{3} y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) + (- (15 x^{4} y^{2}) - (- 30 x^{2} y^{3}) - (45 x y)) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.5

Apply the distributive property.

$\frac{5 x y (60 x^{3} y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Simplify each term.

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Step 7.6.1.1

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

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Step 7.6.1.1.1

Move $x^{3}$ .

$\frac{5 (x^{3} x) y (60 y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.1.2

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

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Step 7.6.1.1.2.1

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

$\frac{5 (x^{3} x^{1}) y (60 y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.1.2.2

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

$\frac{5 x^{3 + 1} y (60 y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.1.3

Add $3$ and $1$ .

$\frac{5 x^{4} y (60 y^{2}) + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.2

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

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Step 7.6.1.2.1

Move $y^{2}$ .

$\frac{5 x^{4} (y^{2} y) \cdot 60 + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.2.2

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

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Step 7.6.1.2.2.1

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

$\frac{5 x^{4} (y^{2} y^{1}) \cdot 60 + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.2.2.2

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

$\frac{5 x^{4} y^{2 + 1} \cdot 60 + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.2.3

Add $2$ and $1$ .

$\frac{5 x^{4} y^{3} \cdot 60 + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.3

Multiply $60$ by $5$ .

$\frac{300 x^{4} y^{3} + 5 x y (- 60 y^{3} x) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.4

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

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Step 7.6.1.4.1

Move $x$ .

$\frac{300 x^{4} y^{3} + 5 (x \cdot x) y (- 60 y^{3}) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.4.2

Multiply $x$ by $x$ .

$\frac{300 x^{4} y^{3} + 5 x^{2} y (- 60 y^{3}) + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.5

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

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Step 7.6.1.5.1

Move $y^{3}$ .

$\frac{300 x^{4} y^{3} + 5 x^{2} (y^{3} y) \cdot - 60 + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.5.2

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

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Step 7.6.1.5.2.1

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

$\frac{300 x^{4} y^{3} + 5 x^{2} (y^{3} y^{1}) \cdot - 60 + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.5.2.2

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

$\frac{300 x^{4} y^{3} + 5 x^{2} y^{3 + 1} \cdot - 60 + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.5.3

Add $3$ and $1$ .

$\frac{300 x^{4} y^{3} + 5 x^{2} y^{4} \cdot - 60 + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.6

Multiply $- 60$ by $5$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 5 x y (45 y) - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.7

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

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Step 7.6.1.7.1

Move $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 5 x (y \cdot y) \cdot 45 - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.7.2

Multiply $y$ by $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 5 x y^{2} \cdot 45 - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.8

Multiply $45$ by $5$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - (15 x^{4} y^{2}) (5 y) - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.9

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

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Step 7.6.1.9.1

Move $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - (15 x^{4} (y \cdot y^{2})) \cdot 5 - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.9.2

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

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Step 7.6.1.9.2.1

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

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - (15 x^{4} (y^{1} y^{2})) \cdot 5 - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.9.2.2

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

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - (15 x^{4} y^{1 + 2}) \cdot 5 - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.9.3

Add $1$ and $2$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - (15 x^{4} y^{3}) \cdot 5 - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.10

Multiply $15$ by $- 1$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 15 (x^{4} y^{3}) \cdot 5 - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.11

Multiply $5$ by $- 15$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} - (- 30 x^{2} y^{3}) (5 y) - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.12

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

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Step 7.6.1.12.1

Move $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} - (- 30 x^{2} (y \cdot y^{3})) \cdot 5 - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.12.2

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

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Step 7.6.1.12.2.1

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

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} - (- 30 x^{2} (y^{1} y^{3})) \cdot 5 - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.12.2.2

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

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} - (- 30 x^{2} y^{1 + 3}) \cdot 5 - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.12.3

Add $1$ and $3$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} - (- 30 x^{2} y^{4}) \cdot 5 - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.13

Multiply $- 30$ by $- 1$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 30 (x^{2} y^{4}) \cdot 5 - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.14

Multiply $5$ by $30$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - (45 x y) (5 y)}{5^{2} x^{2} y^{2}}$

Step 7.6.1.15

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

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Step 7.6.1.15.1

Move $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - (45 x (y \cdot y)) \cdot 5}{5^{2} x^{2} y^{2}}$

Step 7.6.1.15.2

Multiply $y$ by $y$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - (45 x y^{2}) \cdot 5}{5^{2} x^{2} y^{2}}$

Step 7.6.1.16

Multiply $45$ by $- 1$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - 45 (x y^{2}) \cdot 5}{5^{2} x^{2} y^{2}}$

Step 7.6.1.17

Multiply $5$ by $- 45$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - 225 x y^{2}}{5^{2} x^{2} y^{2}}$

Step 7.6.2

Combine the opposite terms in $300 x^{4} y^{3} - 300 x^{2} y^{4} + 225 x y^{2} - 75 x^{4} y^{3} + 150 x^{2} y^{4} - 225 x y^{2}$ .

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Step 7.6.2.1

Subtract $225 x y^{2}$ from $225 x y^{2}$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} - 75 x^{4} y^{3} + 150 x^{2} y^{4} + 0}{5^{2} x^{2} y^{2}}$

Step 7.6.2.2

Add $300 x^{4} y^{3} - 300 x^{2} y^{4} - 75 x^{4} y^{3} + 150 x^{2} y^{4}$ and $0$ .

$\frac{300 x^{4} y^{3} - 300 x^{2} y^{4} - 75 x^{4} y^{3} + 150 x^{2} y^{4}}{5^{2} x^{2} y^{2}}$

Step 7.6.3

Subtract $75 x^{4} y^{3}$ from $300 x^{4} y^{3}$ .

$\frac{225 x^{4} y^{3} - 300 x^{2} y^{4} + 150 x^{2} y^{4}}{5^{2} x^{2} y^{2}}$

Step 7.6.4

Add $- 300 x^{2} y^{4}$ and $150 x^{2} y^{4}$ .

$\frac{225 x^{4} y^{3} - 150 x^{2} y^{4}}{5^{2} x^{2} y^{2}}$

Step 7.7

Raise $5$ to the power of $2$ .

$\frac{225 x^{4} y^{3} - 150 x^{2} y^{4}}{25 x^{2} y^{2}}$

Step 7.8

Reorder terms.

$\frac{225 x^{4} y^{3} - 150 y^{4} x^{2}}{25 x^{2} y^{2}}$

Step 7.9

Factor $75 x^{2} y^{3}$ out of $225 x^{4} y^{3} - 150 y^{4} x^{2}$ .

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Step 7.9.1

Factor $75 x^{2} y^{3}$ out of $225 x^{4} y^{3}$ .

$\frac{75 x^{2} y^{3} (3 x^{2}) - 150 y^{4} x^{2}}{25 x^{2} y^{2}}$

Step 7.9.2

Factor $75 x^{2} y^{3}$ out of $- 150 y^{4} x^{2}$ .

$\frac{75 x^{2} y^{3} (3 x^{2}) + 75 x^{2} y^{3} (- 2 y)}{25 x^{2} y^{2}}$

Step 7.9.3

Factor $75 x^{2} y^{3}$ out of $75 x^{2} y^{3} (3 x^{2}) + 75 x^{2} y^{3} (- 2 y)$ .

$\frac{75 x^{2} y^{3} (3 x^{2} - 2 y)}{25 x^{2} y^{2}}$

Step 7.10

Cancel the common factor of $75$ and $25$ .

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Step 7.10.1

Factor $25$ out of $75 x^{2} y^{3} (3 x^{2} - 2 y)$ .

$\frac{25 (3 x^{2} y^{3} (3 x^{2} - 2 y))}{25 x^{2} y^{2}}$

Step 7.10.2

Cancel the common factors.

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Step 7.10.2.1

Factor $25$ out of $25 x^{2} y^{2}$ .

$\frac{25 (3 x^{2} y^{3} (3 x^{2} - 2 y))}{25 (x^{2} y^{2})}$

Step 7.10.2.2

Cancel the common factor.

$\frac{25 (3 x^{2} y^{3} (3 x^{2} - 2 y))}{25 (x^{2} y^{2})}$

Step 7.10.2.3

Rewrite the expression.

$\frac{3 x^{2} y^{3} (3 x^{2} - 2 y)}{x^{2} y^{2}}$

Step 7.11

Cancel the common factor of $x^{2}$ .

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Step 7.11.1

Cancel the common factor.

$\frac{3 x^{2} y^{3} (3 x^{2} - 2 y)}{x^{2} y^{2}}$

Step 7.11.2

Rewrite the expression.

$\frac{3 y^{3} (3 x^{2} - 2 y)}{y^{2}}$

Step 7.12

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 7.12.1

Factor $y^{2}$ out of $3 y^{3} (3 x^{2} - 2 y)$ .

$\frac{y^{2} (3 y (3 x^{2} - 2 y))}{y^{2}}$

Step 7.12.2

Cancel the common factors.

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Step 7.12.2.1

Multiply by $1$ .

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

Step 7.12.2.2

Cancel the common factor.

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

Step 7.12.2.3

Rewrite the expression.

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

Step 7.12.2.4

Divide $3 y (3 x^{2} - 2 y)$ by $1$ .

$3 y (3 x^{2} - 2 y)$

Step 7.13

Apply the distributive property.

$3 y (3 x^{2}) + 3 y (- 2 y)$

Step 7.14

Rewrite using the commutative property of multiplication.

$3 \cdot 3 y x^{2} + 3 y (- 2 y)$

Step 7.15

Rewrite using the commutative property of multiplication.

$3 \cdot 3 y x^{2} + 3 \cdot - 2 y \cdot y$

Step 7.16

Simplify each term.

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Step 7.16.1

Multiply $3$ by $3$ .

$9 y x^{2} + 3 \cdot - 2 y \cdot y$

Step 7.16.2

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

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Step 7.16.2.1

Move $y$ .

$9 y x^{2} + 3 \cdot - 2 (y \cdot y)$

Step 7.16.2.2

Multiply $y$ by $y$ .

$9 y x^{2} + 3 \cdot - 2 y^{2}$

Step 7.16.3

Multiply $3$ by $- 2$ .

$9 y x^{2} - 6 y^{2}$