Calculus Examples

$(- 10 + 9 x^{- 3} + x^{2}) (7 - 10 x^{2})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 10 + 9 x^{- 3} + x^{2}$ and $g (x) = 7 - 10 x^{2}$ .

$(- 10 + 9 x^{- 3} + x^{2}) \frac{d}{d x} [7 - 10 x^{2}] + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $7 - 10 x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 10 x^{2}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) (\frac{d}{d x} [7] + \frac{d}{d x} [- 10 x^{2}]) + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$(- 10 + 9 x^{- 3} + x^{2}) (0 + \frac{d}{d x} [- 10 x^{2}]) + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 10 x^{2}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) \frac{d}{d x} [- 10 x^{2}] + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x^{2}$ with respect to $x$ is $- 10 \frac{d}{d x} [x^{2}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 10 \frac{d}{d x} [x^{2}]) + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.5

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

$(- 10 + 9 x^{- 3} + x^{2}) (- 10 (2 x)) + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.6

Multiply $2$ by $- 10$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) \frac{d}{d x} [- 10 + 9 x^{- 3} + x^{2}]$

Step 2.7

By the Sum Rule, the derivative of $- 10 + 9 x^{- 3} + x^{2}$ with respect to $x$ is $\frac{d}{d x} [- 10] + \frac{d}{d x} [9 x^{- 3}] + \frac{d}{d x} [x^{2}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (\frac{d}{d x} [- 10] + \frac{d}{d x} [9 x^{- 3}] + \frac{d}{d x} [x^{2}])$

Step 2.8

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (0 + \frac{d}{d x} [9 x^{- 3}] + \frac{d}{d x} [x^{2}])$

Step 2.9

Add $0$ and $\frac{d}{d x} [9 x^{- 3}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (\frac{d}{d x} [9 x^{- 3}] + \frac{d}{d x} [x^{2}])$

Step 2.10

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{- 3}$ with respect to $x$ is $9 \frac{d}{d x} [x^{- 3}]$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (9 \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [x^{2}])$

Step 2.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (9 (- 3 x^{- 4}) + \frac{d}{d x} [x^{2}])$

Step 2.12

Multiply $- 3$ by $9$ .

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (- 27 x^{- 4} + \frac{d}{d x} [x^{2}])$

Step 2.13

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

$(- 10 + 9 x^{- 3} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (- 27 x^{- 4} + 2 x)$

Step 3

Simplify.

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Step 3.1

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

$(- 10 + 9 \frac{1}{x^{3}} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (- 27 x^{- 4} + 2 x)$

Step 3.2

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

$(- 10 + 9 \frac{1}{x^{3}} + x^{2}) (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.3

Apply the distributive property.

$- 10 (- 20 x) + 9 \frac{1}{x^{3}} (- 20 x) + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4

Combine terms.

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Step 3.4.1

Multiply $- 20$ by $- 10$ .

$200 x + 9 \frac{1}{x^{3}} (- 20 x) + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.2

Combine $9$ and $\frac{1}{x^{3}}$ .

$200 x + \frac{9}{x^{3}} (- 20 x) + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.3

Combine $- 20$ and $\frac{9}{x^{3}}$ .

$200 x + \frac{- 20 \cdot 9}{x^{3}} x + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.4

Multiply $- 20$ by $9$ .

$200 x + \frac{- 180}{x^{3}} x + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.5

Combine $\frac{- 180}{x^{3}}$ and $x$ .

$200 x + \frac{- 180 x}{x^{3}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.6

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

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Step 3.4.6.1

Factor $x$ out of $- 180 x$ .

$200 x + \frac{x \cdot - 180}{x^{3}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.6.2

Cancel the common factors.

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Step 3.4.6.2.1

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

$200 x + \frac{x \cdot - 180}{x \cdot x^{2}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.6.2.2

Cancel the common factor.

$200 x + \frac{x \cdot - 180}{x \cdot x^{2}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.6.2.3

Rewrite the expression.

$200 x + \frac{- 180}{x^{2}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.7

Move the negative in front of the fraction.

$200 x - \frac{180}{x^{2}} + x^{2} (- 20 x) + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.8

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

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Step 3.4.8.1

Move $x$ .

$200 x - \frac{180}{x^{2}} + x \cdot x^{2} \cdot - 20 + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.8.2

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

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Step 3.4.8.2.1

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

$200 x - \frac{180}{x^{2}} + x^{1} x^{2} \cdot - 20 + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.8.2.2

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

$200 x - \frac{180}{x^{2}} + x^{1 + 2} \cdot - 20 + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.8.3

Add $1$ and $2$ .

$200 x - \frac{180}{x^{2}} + x^{3} \cdot - 20 + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.9

Move $- 20$ to the left of $x^{3}$ .

$200 x - \frac{180}{x^{2}} - 20 \cdot x^{3} + (7 - 10 x^{2}) (- 27 \frac{1}{x^{4}} + 2 x)$

Step 3.4.10

Combine $- 27$ and $\frac{1}{x^{4}}$ .

$200 x - \frac{180}{x^{2}} - 20 x^{3} + (7 - 10 x^{2}) (\frac{- 27}{x^{4}} + 2 x)$

Step 3.4.11

Move the negative in front of the fraction.

$200 x - \frac{180}{x^{2}} - 20 x^{3} + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x)$

Step 3.4.12

To write $(7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x)$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$200 x - 20 x^{3} - \frac{180}{x^{2}} + \frac{(7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.13

Combine the numerators over the common denominator.

$200 x - 20 x^{3} + \frac{- 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.14

To write $200 x$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$- 20 x^{3} + \frac{200 x \cdot x^{2}}{x^{2}} + \frac{- 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.15

Combine the numerators over the common denominator.

$- 20 x^{3} + \frac{200 x \cdot x^{2} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.16

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

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Step 3.4.16.1

Move $x^{2}$ .

$- 20 x^{3} + \frac{200 (x^{2} x) - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.16.2

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

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Step 3.4.16.2.1

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

$- 20 x^{3} + \frac{200 (x^{2} x^{1}) - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.16.2.2

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

$- 20 x^{3} + \frac{200 x^{2 + 1} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.16.3

Add $2$ and $1$ .

$- 20 x^{3} + \frac{200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.17

To write $- 20 x^{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$- 20 x^{3} \cdot \frac{x^{2}}{x^{2}} + \frac{200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.18

Combine $- 20 x^{3}$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- 20 x^{3} x^{2}}{x^{2}} + \frac{200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.19

Combine the numerators over the common denominator.

$\frac{- 20 x^{3} x^{2} + 200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.20

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.4.20.1

Move $x^{2}$ .

$\frac{- 20 (x^{2} x^{3}) + 200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.20.2

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

$\frac{- 20 x^{2 + 3} + 200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.4.20.3

Add $2$ and $3$ .

$\frac{- 20 x^{5} + 200 x^{3} - 180 + (7 - 10 x^{2}) (- \frac{27}{x^{4}} + 2 x) x^{2}}{x^{2}}$

Step 3.5

Reorder terms.

$\frac{- 20 x^{5} + 200 x^{3} + x^{2} (- \frac{27}{x^{4}} + 2 x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{- 20 x^{5} + 200 x^{3} + (x^{2} (- \frac{27}{x^{4}}) + x^{2} (2 x)) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.2

Rewrite using the commutative property of multiplication.

$\frac{- 20 x^{5} + 200 x^{3} + (- x^{2} \frac{27}{x^{4}} + x^{2} (2 x)) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.3

Rewrite using the commutative property of multiplication.

$\frac{- 20 x^{5} + 200 x^{3} + (- x^{2} \frac{27}{x^{4}} + 2 x^{2} x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4

Simplify each term.

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Step 3.6.4.1

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

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Step 3.6.4.1.1

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

$\frac{- 20 x^{5} + 200 x^{3} + (x^{2} \cdot - 1 \frac{27}{x^{4}} + 2 x^{2} x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.1.2

Factor $x^{2}$ out of $x^{4}$ .

$\frac{- 20 x^{5} + 200 x^{3} + (x^{2} \cdot - 1 \frac{27}{x^{2} x^{2}} + 2 x^{2} x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.1.3

Cancel the common factor.

$\frac{- 20 x^{5} + 200 x^{3} + (x^{2} \cdot - 1 \frac{27}{x^{2} x^{2}} + 2 x^{2} x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.1.4

Rewrite the expression.

$\frac{- 20 x^{5} + 200 x^{3} + (- \frac{27}{x^{2}} + 2 x^{2} x) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.2

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

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Step 3.6.4.2.1

Move $x$ .

$\frac{- 20 x^{5} + 200 x^{3} + (- \frac{27}{x^{2}} + 2 (x \cdot x^{2})) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.2.2

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

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Step 3.6.4.2.2.1

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

$\frac{- 20 x^{5} + 200 x^{3} + (- \frac{27}{x^{2}} + 2 (x^{1} x^{2})) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.2.2.2

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

$\frac{- 20 x^{5} + 200 x^{3} + (- \frac{27}{x^{2}} + 2 x^{1 + 2}) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.4.2.3

Add $1$ and $2$ .

$\frac{- 20 x^{5} + 200 x^{3} + (- \frac{27}{x^{2}} + 2 x^{3}) (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.5

Expand $(- \frac{27}{x^{2}} + 2 x^{3}) (7 - 10 x^{2})$ using the FOIL Method.

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Step 3.6.5.1

Apply the distributive property.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{27}{x^{2}} (7 - 10 x^{2}) + 2 x^{3} (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.5.2

Apply the distributive property.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{27}{x^{2}} \cdot 7 - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} (7 - 10 x^{2}) - 180}{x^{2}}$

Step 3.6.5.3

Apply the distributive property.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{27}{x^{2}} \cdot 7 - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6

Simplify each term.

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Step 3.6.6.1

Multiply $- \frac{27}{x^{2}} \cdot 7$ .

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Step 3.6.6.1.1

Multiply $7$ by $- 1$ .

$\frac{- 20 x^{5} + 200 x^{3} - 7 \frac{27}{x^{2}} - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.1.2

Combine $- 7$ and $\frac{27}{x^{2}}$ .

$\frac{- 20 x^{5} + 200 x^{3} + \frac{- 7 \cdot 27}{x^{2}} - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.1.3

Multiply $- 7$ by $27$ .

$\frac{- 20 x^{5} + 200 x^{3} + \frac{- 189}{x^{2}} - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.2

Move the negative in front of the fraction.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} - \frac{27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.3

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

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Step 3.6.6.3.1

Move the leading negative in $- \frac{27}{x^{2}}$ into the numerator.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + \frac{- 27}{x^{2}} (- 10 x^{2}) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.3.2

Factor $x^{2}$ out of $- 10 x^{2}$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + \frac{- 27}{x^{2}} (x^{2} \cdot - 10) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.3.3

Cancel the common factor.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + \frac{- 27}{x^{2}} (x^{2} \cdot - 10) + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.3.4

Rewrite the expression.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} - 27 \cdot - 10 + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.4

Multiply $- 27$ by $- 10$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 2 x^{3} \cdot 7 + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.5

Multiply $7$ by $2$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} + 2 x^{3} (- 10 x^{2}) - 180}{x^{2}}$

Step 3.6.6.6

Rewrite using the commutative property of multiplication.

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} + 2 \cdot - 10 x^{3} x^{2} - 180}{x^{2}}$

Step 3.6.6.7

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.6.6.7.1

Move $x^{2}$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} + 2 \cdot - 10 (x^{2} x^{3}) - 180}{x^{2}}$

Step 3.6.6.7.2

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

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} + 2 \cdot - 10 x^{2 + 3} - 180}{x^{2}}$

Step 3.6.6.7.3

Add $2$ and $3$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} + 2 \cdot - 10 x^{5} - 180}{x^{2}}$

Step 3.6.6.8

Multiply $2$ by $- 10$ .

$\frac{- 20 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} - 20 x^{5} - 180}{x^{2}}$

Step 3.6.7

Subtract $20 x^{5}$ from $- 20 x^{5}$ .

$\frac{- 40 x^{5} + 200 x^{3} - \frac{189}{x^{2}} + 270 + 14 x^{3} - 180}{x^{2}}$

Step 3.6.8

Add $200 x^{3}$ and $14 x^{3}$ .

$\frac{- 40 x^{5} + 214 x^{3} - \frac{189}{x^{2}} + 270 - 180}{x^{2}}$

Step 3.6.9

Subtract $180$ from $270$ .

$\frac{- 40 x^{5} + 214 x^{3} - \frac{189}{x^{2}} + 90}{x^{2}}$

Step 3.6.10

To write $- 40 x^{5}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{214 x^{3} - 40 x^{5} \cdot \frac{x^{2}}{x^{2}} - \frac{189}{x^{2}} + 90}{x^{2}}$

Step 3.6.11

Combine $- 40 x^{5}$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{214 x^{3} + \frac{- 40 x^{5} x^{2}}{x^{2}} - \frac{189}{x^{2}} + 90}{x^{2}}$

Step 3.6.12

Combine the numerators over the common denominator.

$\frac{214 x^{3} + \frac{- 40 x^{5} x^{2} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.13

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 3.6.13.1

Move $x^{2}$ .

$\frac{214 x^{3} + \frac{- 40 (x^{2} x^{5}) - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.13.2

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

$\frac{214 x^{3} + \frac{- 40 x^{2 + 5} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.13.3

Add $2$ and $5$ .

$\frac{214 x^{3} + \frac{- 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.14

To write $214 x^{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{\frac{214 x^{3} x^{2}}{x^{2}} + \frac{- 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.15

Combine the numerators over the common denominator.

$\frac{\frac{214 x^{3} x^{2} - 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.16

Simplify the numerator.

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Step 3.6.16.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.6.16.1.1

Move $x^{2}$ .

$\frac{\frac{214 (x^{2} x^{3}) - 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.16.1.2

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

$\frac{\frac{214 x^{2 + 3} - 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.16.1.3

Add $2$ and $3$ .

$\frac{\frac{214 x^{5} - 40 x^{7} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.16.2

Reorder terms.

$\frac{\frac{- 40 x^{7} + 214 x^{5} - 189}{x^{2}} + 90}{x^{2}}$

Step 3.6.17

To write $90$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{\frac{- 40 x^{7} + 214 x^{5} - 189}{x^{2}} + \frac{90 x^{2}}{x^{2}}}{x^{2}}$

Step 3.6.18

Combine the numerators over the common denominator.

$\frac{\frac{- 40 x^{7} + 214 x^{5} - 189 + 90 x^{2}}{x^{2}}}{x^{2}}$

Step 3.6.19

Reorder terms.

$\frac{\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2}}}{x^{2}}$

Step 3.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2}} \cdot \frac{1}{x^{2}}$

Step 3.8

Multiply $\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2}} \cdot \frac{1}{x^{2}}$ .

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Step 3.8.1

Multiply $\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2}}$ by $\frac{1}{x^{2}}$ .

$\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2} x^{2}}$

Step 3.8.2

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

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Step 3.8.2.1

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

$\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{2 + 2}}$

Step 3.8.2.2

Add $2$ and $2$ .

$\frac{- 40 x^{7} + 214 x^{5} + 90 x^{2} - 189}{x^{4}}$

Step 3.9

Factor $- 1$ out of $- 40 x^{7}$ .

$\frac{- (40 x^{7}) + 214 x^{5} + 90 x^{2} - 189}{x^{4}}$

Step 3.10

Factor $- 1$ out of $214 x^{5}$ .

$\frac{- (40 x^{7}) - (- 214 x^{5}) + 90 x^{2} - 189}{x^{4}}$

Step 3.11

Factor $- 1$ out of $- (40 x^{7}) - (- 214 x^{5})$ .

$\frac{- (40 x^{7} - 214 x^{5}) + 90 x^{2} - 189}{x^{4}}$

Step 3.12

Factor $- 1$ out of $90 x^{2}$ .

$\frac{- (40 x^{7} - 214 x^{5}) - (- 90 x^{2}) - 189}{x^{4}}$

Step 3.13

Factor $- 1$ out of $- (40 x^{7} - 214 x^{5}) - (- 90 x^{2})$ .

$\frac{- (40 x^{7} - 214 x^{5} - 90 x^{2}) - 189}{x^{4}}$

Step 3.14

Rewrite $- 189$ as $- 1 (189)$ .

$\frac{- (40 x^{7} - 214 x^{5} - 90 x^{2}) - 1 (189)}{x^{4}}$

Step 3.15

Factor $- 1$ out of $- (40 x^{7} - 214 x^{5} - 90 x^{2}) - 1 (189)$ .

$\frac{- (40 x^{7} - 214 x^{5} - 90 x^{2} + 189)}{x^{4}}$

Step 3.16

Rewrite $- (40 x^{7} - 214 x^{5} - 90 x^{2} + 189)$ as $- 1 (40 x^{7} - 214 x^{5} - 90 x^{2} + 189)$ .

$\frac{- 1 (40 x^{7} - 214 x^{5} - 90 x^{2} + 189)}{x^{4}}$

Step 3.17

Move the negative in front of the fraction.

$- \frac{40 x^{7} - 214 x^{5} - 90 x^{2} + 189}{x^{4}}$