Algebra Examples

$\frac{8 y^{3}}{2 y + 7} + \frac{5 y}{y + 6}$

Step 1

To write $\frac{8 y^{3}}{2 y + 7}$ as a fraction with a common denominator, multiply by $\frac{y + 6}{y + 6}$ .

$\frac{8 y^{3}}{2 y + 7} \cdot \frac{y + 6}{y + 6} + \frac{5 y}{y + 6}$

Step 2

To write $\frac{5 y}{y + 6}$ as a fraction with a common denominator, multiply by $\frac{2 y + 7}{2 y + 7}$ .

$\frac{8 y^{3}}{2 y + 7} \cdot \frac{y + 6}{y + 6} + \frac{5 y}{y + 6} \cdot \frac{2 y + 7}{2 y + 7}$

Step 3

Write each expression with a common denominator of $(2 y + 7) (y + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8 y^{3}}{2 y + 7}$ by $\frac{y + 6}{y + 6}$ .

$\frac{8 y^{3} (y + 6)}{(2 y + 7) (y + 6)} + \frac{5 y}{y + 6} \cdot \frac{2 y + 7}{2 y + 7}$

Step 3.2

Multiply $\frac{5 y}{y + 6}$ by $\frac{2 y + 7}{2 y + 7}$ .

$\frac{8 y^{3} (y + 6)}{(2 y + 7) (y + 6)} + \frac{5 y (2 y + 7)}{(y + 6) (2 y + 7)}$

Step 3.3

Reorder the factors of $(y + 6) (2 y + 7)$ .

$\frac{8 y^{3} (y + 6)}{(2 y + 7) (y + 6)} + \frac{5 y (2 y + 7)}{(2 y + 7) (y + 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 y^{3} (y + 6) + 5 y (2 y + 7)}{(2 y + 7) (y + 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $y$ out of $8 y^{3} (y + 6) + 5 y (2 y + 7)$ .

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Step 5.1.1

Factor $y$ out of $8 y^{3} (y + 6)$ .

$\frac{y (8 y^{2} (y + 6)) + 5 y (2 y + 7)}{(2 y + 7) (y + 6)}$

Step 5.1.2

Factor $y$ out of $5 y (2 y + 7)$ .

$\frac{y (8 y^{2} (y + 6)) + y (5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.1.3

Factor $y$ out of $y (8 y^{2} (y + 6)) + y (5 (2 y + 7))$ .

$\frac{y (8 y^{2} (y + 6) + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.2

Apply the distributive property.

$\frac{y (8 y^{2} y + 8 y^{2} \cdot 6 + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.3

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

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Step 5.3.1

Move $y$ .

$\frac{y (8 (y \cdot y^{2}) + 8 y^{2} \cdot 6 + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.3.2

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

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Step 5.3.2.1

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

$\frac{y (8 (y^{1} y^{2}) + 8 y^{2} \cdot 6 + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.3.2.2

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

$\frac{y (8 y^{1 + 2} + 8 y^{2} \cdot 6 + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.3.3

Add $1$ and $2$ .

$\frac{y (8 y^{3} + 8 y^{2} \cdot 6 + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.4

Multiply $6$ by $8$ .

$\frac{y (8 y^{3} + 48 y^{2} + 5 (2 y + 7))}{(2 y + 7) (y + 6)}$

Step 5.5

Apply the distributive property.

$\frac{y (8 y^{3} + 48 y^{2} + 5 (2 y) + 5 \cdot 7)}{(2 y + 7) (y + 6)}$

Step 5.6

Multiply $2$ by $5$ .

$\frac{y (8 y^{3} + 48 y^{2} + 10 y + 5 \cdot 7)}{(2 y + 7) (y + 6)}$

Step 5.7

Multiply $5$ by $7$ .

$\frac{y (8 y^{3} + 48 y^{2} + 10 y + 35)}{(2 y + 7) (y + 6)}$