Calculus Examples

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (y + 9 y^{3})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = \frac{1}{y^{2}} - \frac{3}{y^{4}}$ and $g (y) = y + 9 y^{3}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) \frac{d}{d y} [y + 9 y^{3}] + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2

Differentiate.

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Step 2.1

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (\frac{d}{d y} [y] + \frac{d}{d y} [9 y^{3}]) + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2.2

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + \frac{d}{d y} [9 y^{3}]) + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2.3

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 9 \frac{d}{d y} [y^{3}]) + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2.4

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 9 (3 y^{2})) + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2.5

Multiply $3$ by $9$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) \frac{d}{d y} [\frac{1}{y^{2}} - \frac{3}{y^{4}}]$

Step 2.6

By the Sum Rule, the derivative of $\frac{1}{y^{2}} - \frac{3}{y^{4}}$ with respect to $y$ is $\frac{d}{d y} [\frac{1}{y^{2}}] + \frac{d}{d y} [- \frac{3}{y^{4}}]$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (\frac{d}{d y} [\frac{1}{y^{2}}] + \frac{d}{d y} [- \frac{3}{y^{4}}])$

Step 2.7

Apply basic rules of exponents.

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Step 2.7.1

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (\frac{d}{d y} [{(y^{2})}^{- 1}] + \frac{d}{d y} [- \frac{3}{y^{4}}])$

Step 2.7.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

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Step 2.7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (\frac{d}{d y} [y^{2 \cdot - 1}] + \frac{d}{d y} [- \frac{3}{y^{4}}])$

Step 2.7.2.2

Multiply $2$ by $- 1$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (\frac{d}{d y} [y^{- 2}] + \frac{d}{d y} [- \frac{3}{y^{4}}])$

Step 2.8

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} + \frac{d}{d y} [- \frac{3}{y^{4}}])$

Step 2.9

Since $- 3$ is constant with respect to $y$ , the derivative of $- \frac{3}{y^{4}}$ with respect to $y$ is $- 3 \frac{d}{d y} [\frac{1}{y^{4}}]$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} - 3 \frac{d}{d y} [\frac{1}{y^{4}}])$

Step 2.10

Apply basic rules of exponents.

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Step 2.10.1

Rewrite $\frac{1}{y^{4}}$ as ${(y^{4})}^{- 1}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} - 3 \frac{d}{d y} [{(y^{4})}^{- 1}])$

Step 2.10.2

Multiply the exponents in ${(y^{4})}^{- 1}$ .

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Step 2.10.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} - 3 \frac{d}{d y} [y^{4 \cdot - 1}])$

Step 2.10.2.2

Multiply $4$ by $- 1$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} - 3 \frac{d}{d y} [y^{- 4}])$

Step 2.11

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} - 3 (- 4 y^{- 5}))$

Step 2.12

Multiply $- 4$ by $- 3$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 y^{- 3} + 12 y^{- 5})$

Step 3

Simplify.

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Step 3.1

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 \frac{1}{y^{3}} + 12 y^{- 5})$

Step 3.2

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

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- 2 \frac{1}{y^{3}} + 12 \frac{1}{y^{5}})$

Step 3.3

Combine terms.

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Step 3.3.1

Combine $- 2$ and $\frac{1}{y^{3}}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (\frac{- 2}{y^{3}} + 12 \frac{1}{y^{5}})$

Step 3.3.2

Move the negative in front of the fraction.

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- \frac{2}{y^{3}} + 12 \frac{1}{y^{5}})$

Step 3.3.3

Combine $12$ and $\frac{1}{y^{5}}$ .

$(\frac{1}{y^{2}} - \frac{3}{y^{4}}) (1 + 27 y^{2}) + (y + 9 y^{3}) (- \frac{2}{y^{3}} + \frac{12}{y^{5}})$

Step 3.4

Reorder terms.

$(1 + 27 y^{2}) (\frac{1}{y^{2}} - \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5

Simplify each term.

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Step 3.5.1

Expand $(1 + 27 y^{2}) (\frac{1}{y^{2}} - \frac{3}{y^{4}})$ using the FOIL Method.

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Step 3.5.1.1

Apply the distributive property.

$1 (\frac{1}{y^{2}} - \frac{3}{y^{4}}) + 27 y^{2} (\frac{1}{y^{2}} - \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.1.2

Apply the distributive property.

$1 \frac{1}{y^{2}} + 1 (- \frac{3}{y^{4}}) + 27 y^{2} (\frac{1}{y^{2}} - \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.1.3

Apply the distributive property.

$1 \frac{1}{y^{2}} + 1 (- \frac{3}{y^{4}}) + 27 y^{2} \frac{1}{y^{2}} + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2

Simplify and combine like terms.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Multiply $\frac{1}{y^{2}}$ by $1$ .

$\frac{1}{y^{2}} + 1 (- \frac{3}{y^{4}}) + 27 y^{2} \frac{1}{y^{2}} + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.2

Multiply $- \frac{3}{y^{4}}$ by $1$ .

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 y^{2} \frac{1}{y^{2}} + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.3

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

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Step 3.5.2.1.3.1

Factor $y^{2}$ out of $27 y^{2}$ .

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + y^{2} \cdot 27 \frac{1}{y^{2}} + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.3.2

Cancel the common factor.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + y^{2} \cdot 27 \frac{1}{y^{2}} + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.3.3

Rewrite the expression.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + 27 y^{2} (- \frac{3}{y^{4}}) + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.4

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

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Step 3.5.2.1.4.1

Move the leading negative in $- \frac{3}{y^{4}}$ into the numerator.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + 27 y^{2} \frac{- 3}{y^{4}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.4.2

Factor $y^{2}$ out of $27 y^{2}$ .

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + y^{2} \cdot 27 \frac{- 3}{y^{4}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.4.3

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

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + y^{2} \cdot 27 \frac{- 3}{y^{2} y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.4.4

Cancel the common factor.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + y^{2} \cdot 27 \frac{- 3}{y^{2} y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.4.5

Rewrite the expression.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + 27 \frac{- 3}{y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.5

Combine $27$ and $\frac{- 3}{y^{2}}$ .

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{27 \cdot - 3}{y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.6

Multiply $27$ by $- 3$ .

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 81}{y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.1.7

Move the negative in front of the fraction.

$\frac{1}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{81}{y^{2}} + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.2

Combine the numerators over the common denominator.

$\frac{1 - 81}{y^{2}} - \frac{3}{y^{4}} + 27 + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.2.3

Subtract $81$ from $1$ .

$\frac{- 80}{y^{2}} - \frac{3}{y^{4}} + 27 + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + (- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$

Step 3.5.4

Expand $(- \frac{2}{y^{3}} + \frac{12}{y^{5}}) (y + 9 y^{3})$ using the FOIL Method.

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Step 3.5.4.1

Apply the distributive property.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{3}} (y + 9 y^{3}) + \frac{12}{y^{5}} (y + 9 y^{3})$

Step 3.5.4.2

Apply the distributive property.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{3}} y - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} (y + 9 y^{3})$

Step 3.5.4.3

Apply the distributive property.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{3}} y - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5

Simplify and combine like terms.

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Step 3.5.5.1

Simplify each term.

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Step 3.5.5.1.1

Cancel the common factor of $y$ .

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Step 3.5.5.1.1.1

Move the leading negative in $- \frac{2}{y^{3}}$ into the numerator.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 2}{y^{3}} y - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.1.2

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

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 2}{y \cdot y^{2}} y - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.1.3

Cancel the common factor.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 2}{y \cdot y^{2}} y - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.1.4

Rewrite the expression.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 2}{y^{2}} - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.2

Move the negative in front of the fraction.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - \frac{2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.3

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

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Step 3.5.5.1.3.1

Move the leading negative in $- \frac{2}{y^{3}}$ into the numerator.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (9 y^{3}) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.3.2

Factor $y^{3}$ out of $9 y^{3}$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (y^{3} \cdot 9) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.3.3

Cancel the common factor.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} + \frac{- 2}{y^{3}} (y^{3} \cdot 9) + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.3.4

Rewrite the expression.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 2 \cdot 9 + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.4

Multiply $- 2$ by $9$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{5}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.5

Cancel the common factor of $y$ .

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Step 3.5.5.1.5.1

Factor $y$ out of $y^{5}$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y \cdot y^{4}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.5.2

Cancel the common factor.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y \cdot y^{4}} y + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.5.3

Rewrite the expression.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{12}{y^{5}} (9 y^{3})$

Step 3.5.5.1.6

Rewrite using the commutative property of multiplication.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + 9 \frac{12}{y^{5}} y^{3}$

Step 3.5.5.1.7

Multiply $9 \frac{12}{y^{5}}$ .

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Step 3.5.5.1.7.1

Combine $9$ and $\frac{12}{y^{5}}$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{9 \cdot 12}{y^{5}} y^{3}$

Step 3.5.5.1.7.2

Multiply $9$ by $12$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{108}{y^{5}} y^{3}$

Step 3.5.5.1.8

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

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Step 3.5.5.1.8.1

Factor $y^{3}$ out of $y^{5}$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{108}{y^{3} y^{2}} y^{3}$

Step 3.5.5.1.8.2

Cancel the common factor.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{108}{y^{3} y^{2}} y^{3}$

Step 3.5.5.1.8.3

Rewrite the expression.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 - \frac{2}{y^{2}} - 18 + \frac{12}{y^{4}} + \frac{108}{y^{2}}$

Step 3.5.5.2

Combine the numerators over the common denominator.

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{- 2 + 108}{y^{2}} - 18 + \frac{12}{y^{4}}$

Step 3.5.5.3

Add $- 2$ and $108$ .

$- \frac{80}{y^{2}} - \frac{3}{y^{4}} + 27 + \frac{106}{y^{2}} - 18 + \frac{12}{y^{4}}$

Step 3.6

Combine the numerators over the common denominator.

$27 - 18 + \frac{- 80 + 106}{y^{2}} + \frac{- 3 + 12}{y^{4}}$

Step 3.7

Add $- 80$ and $106$ .

$27 - 18 + \frac{26}{y^{2}} + \frac{- 3 + 12}{y^{4}}$

Step 3.8

Add $- 3$ and $12$ .

$27 - 18 + \frac{26}{y^{2}} + \frac{9}{y^{4}}$

Step 3.9

Subtract $18$ from $27$ .

$9 + \frac{26}{y^{2}} + \frac{9}{y^{4}}$