Basic Math Examples

$\frac{8}{3 y (y + 1)} + \frac{y - 3}{y^{2} (y - 2)}$

Step 1

To write $\frac{8}{3 y (y + 1)}$ as a fraction with a common denominator, multiply by $\frac{(y - 2) y}{(y - 2) y}$ .

$\frac{8}{3 y (y + 1)} \cdot \frac{(y - 2) y}{(y - 2) y} + \frac{y - 3}{y^{2} (y - 2)}$

Step 2

To write $\frac{y - 3}{y^{2} (y - 2)}$ as a fraction with a common denominator, multiply by $\frac{3 (y + 1)}{3 (y + 1)}$ .

$\frac{8}{3 y (y + 1)} \cdot \frac{(y - 2) y}{(y - 2) y} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3

Write each expression with a common denominator of $3 (y + 1) (y - 2) y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8}{3 y (y + 1)}$ by $\frac{(y - 2) y}{(y - 2) y}$ .

$\frac{8 ((y - 2) y)}{3 y (y + 1) ((y - 2) y)} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3.2

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

$\frac{8 ((y - 2) y)}{3 (y^{1} y) (y + 1) (y - 2)} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3.3

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

$\frac{8 ((y - 2) y)}{3 (y^{1} y^{1}) (y + 1) (y - 2)} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3.4

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

$\frac{8 ((y - 2) y)}{3 y^{1 + 1} (y + 1) (y - 2)} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3.5

Add $1$ and $1$ .

$\frac{8 ((y - 2) y)}{3 y^{2} (y + 1) (y - 2)} + \frac{y - 3}{y^{2} (y - 2)} \cdot \frac{3 (y + 1)}{3 (y + 1)}$

Step 3.6

Multiply $\frac{y - 3}{y^{2} (y - 2)}$ by $\frac{3 (y + 1)}{3 (y + 1)}$ .

$\frac{8 ((y - 2) y)}{3 y^{2} (y + 1) (y - 2)} + \frac{(y - 3) (3 (y + 1))}{y^{2} (y - 2) (3 (y + 1))}$

Step 3.7

Reorder the factors of $y^{2} (y - 2) (3 (y + 1))$ .

$\frac{8 ((y - 2) y)}{3 y^{2} (y + 1) (y - 2)} + \frac{(y - 3) (3 (y + 1))}{3 y^{2} (y + 1) (y - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 ((y - 2) y) + (y - 3) (3 (y + 1))}{3 y^{2} (y + 1) (y - 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(8 y + 8 \cdot - 2) y + (y - 3) \cdot 3 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.2

Multiply $8$ by $- 2$ .

$\frac{(8 y - 16) y + (y - 3) \cdot 3 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.3

Apply the distributive property.

$\frac{8 y \cdot y - 16 y + (y - 3) \cdot 3 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.4

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

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Step 5.4.1

Move $y$ .

$\frac{8 (y \cdot y) - 16 y + (y - 3) \cdot 3 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.4.2

Multiply $y$ by $y$ .

$\frac{8 y^{2} - 16 y + (y - 3) \cdot 3 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.5

Apply the distributive property.

$\frac{8 y^{2} - 16 y + (y \cdot 3 - 3 \cdot 3) (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.6

Move $3$ to the left of $y$ .

$\frac{8 y^{2} - 16 y + (3 \cdot y - 3 \cdot 3) (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.7

Multiply $- 3$ by $3$ .

$\frac{8 y^{2} - 16 y + (3 \cdot y - 9) (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.8

Expand $(3 y - 9) (y + 1)$ using the FOIL Method.

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Step 5.8.1

Apply the distributive property.

$\frac{8 y^{2} - 16 y + 3 y (y + 1) - 9 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.8.2

Apply the distributive property.

$\frac{8 y^{2} - 16 y + 3 y \cdot y + 3 y \cdot 1 - 9 (y + 1)}{3 y^{2} (y + 1) (y - 2)}$

Step 5.8.3

Apply the distributive property.

$\frac{8 y^{2} - 16 y + 3 y \cdot y + 3 y \cdot 1 - 9 y - 9 \cdot 1}{3 y^{2} (y + 1) (y - 2)}$

Step 5.9

Simplify and combine like terms.

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Step 5.9.1

Simplify each term.

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Step 5.9.1.1

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

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Step 5.9.1.1.1

Move $y$ .

$\frac{8 y^{2} - 16 y + 3 (y \cdot y) + 3 y \cdot 1 - 9 y - 9 \cdot 1}{3 y^{2} (y + 1) (y - 2)}$

Step 5.9.1.1.2

Multiply $y$ by $y$ .

$\frac{8 y^{2} - 16 y + 3 y^{2} + 3 y \cdot 1 - 9 y - 9 \cdot 1}{3 y^{2} (y + 1) (y - 2)}$

Step 5.9.1.2

Multiply $3$ by $1$ .

$\frac{8 y^{2} - 16 y + 3 y^{2} + 3 y - 9 y - 9 \cdot 1}{3 y^{2} (y + 1) (y - 2)}$

Step 5.9.1.3

Multiply $- 9$ by $1$ .

$\frac{8 y^{2} - 16 y + 3 y^{2} + 3 y - 9 y - 9}{3 y^{2} (y + 1) (y - 2)}$

Step 5.9.2

Subtract $9 y$ from $3 y$ .

$\frac{8 y^{2} - 16 y + 3 y^{2} - 6 y - 9}{3 y^{2} (y + 1) (y - 2)}$

Step 5.10

Add $8 y^{2}$ and $3 y^{2}$ .

$\frac{11 y^{2} - 16 y - 6 y - 9}{3 y^{2} (y + 1) (y - 2)}$

Step 5.11

Subtract $6 y$ from $- 16 y$ .

$\frac{11 y^{2} - 22 y - 9}{3 y^{2} (y + 1) (y - 2)}$