Basic Math Examples

$\frac{8 y + 15}{y^{2 + 5 y}} + \frac{y}{y + 5}$

Step 1

To write $\frac{8 y + 15}{y^{2 + 5 y}}$ as a fraction with a common denominator, multiply by $\frac{y + 5}{y + 5}$ .

$\frac{8 y + 15}{y^{2 + 5 y}} \cdot \frac{y + 5}{y + 5} + \frac{y}{y + 5}$

Step 2

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

$\frac{8 y + 15}{y^{2 + 5 y}} \cdot \frac{y + 5}{y + 5} + \frac{y}{y + 5} \cdot \frac{y^{2 + 5 y}}{y^{2 + 5 y}}$

Step 3

Write each expression with a common denominator of $y^{2 + 5 y} (y + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8 y + 15}{y^{2 + 5 y}}$ by $\frac{y + 5}{y + 5}$ .

$\frac{(8 y + 15) (y + 5)}{y^{2 + 5 y} (y + 5)} + \frac{y}{y + 5} \cdot \frac{y^{2 + 5 y}}{y^{2 + 5 y}}$

Step 3.2

Multiply $\frac{y}{y + 5}$ by $\frac{y^{2 + 5 y}}{y^{2 + 5 y}}$ .

$\frac{(8 y + 15) (y + 5)}{y^{2 + 5 y} (y + 5)} + \frac{y \cdot y^{2 + 5 y}}{(y + 5) y^{2 + 5 y}}$

Step 3.3

Reorder the factors of $(y + 5) y^{2 + 5 y}$ .

$\frac{(8 y + 15) (y + 5)}{y^{2 + 5 y} (y + 5)} + \frac{y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(8 y + 15) (y + 5) + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(8 y + 15) (y + 5)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{8 y (y + 5) + 15 (y + 5) + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.1.2

Apply the distributive property.

$\frac{8 y \cdot y + 8 y \cdot 5 + 15 (y + 5) + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.1.3

Apply the distributive property.

$\frac{8 y \cdot y + 8 y \cdot 5 + 15 y + 15 \cdot 5 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move $y$ .

$\frac{8 (y \cdot y) + 8 y \cdot 5 + 15 y + 15 \cdot 5 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.2.1.1.2

Multiply $y$ by $y$ .

$\frac{8 y^{2} + 8 y \cdot 5 + 15 y + 15 \cdot 5 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.2.1.2

Multiply $5$ by $8$ .

$\frac{8 y^{2} + 40 y + 15 y + 15 \cdot 5 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.2.1.3

Multiply $15$ by $5$ .

$\frac{8 y^{2} + 40 y + 15 y + 75 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.2.2

Add $40 y$ and $15 y$ .

$\frac{8 y^{2} + 55 y + 75 + y \cdot y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.3

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

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Step 5.3.1

Multiply $y$ by $y^{2 + 5 y}$ .

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Step 5.3.1.1

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

$\frac{8 y^{2} + 55 y + 75 + y^{1} y^{2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.3.1.2

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

$\frac{8 y^{2} + 55 y + 75 + y^{1 + 2 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.3.2

Add $1$ and $2$ .

$\frac{8 y^{2} + 55 y + 75 + y^{3 + 5 y}}{y^{2 + 5 y} (y + 5)}$

Step 5.4

Reorder terms.

$\frac{8 y^{2} + 55 y + y^{3 + 5 y} + 75}{y^{2 + 5 y} (y + 5)}$