Algebra Examples

$\frac{8}{3 y} + \frac{2}{9} - \frac{3}{10 y^{2}}$

Step 1

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

$\frac{8}{3 y} \cdot \frac{3}{3} + \frac{2}{9} - \frac{3}{10 y^{2}}$

Step 2

To write $\frac{2}{9}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{8}{3 y} \cdot \frac{3}{3} + \frac{2}{9} \cdot \frac{y}{y} - \frac{3}{10 y^{2}}$

Step 3

Write each expression with a common denominator of $9 y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{8}{3 y}$ by $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3 y \cdot 3} + \frac{2}{9} \cdot \frac{y}{y} - \frac{3}{10 y^{2}}$

Step 3.2

Multiply $3$ by $3$ .

$\frac{8 \cdot 3}{9 y} + \frac{2}{9} \cdot \frac{y}{y} - \frac{3}{10 y^{2}}$

Step 3.3

Multiply $\frac{2}{9}$ by $\frac{y}{y}$ .

$\frac{8 \cdot 3}{9 y} + \frac{2 y}{9 y} - \frac{3}{10 y^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 + 2 y}{9 y} - \frac{3}{10 y^{2}}$

Step 5

Multiply $8$ by $3$ .

$\frac{24 + 2 y}{9 y} - \frac{3}{10 y^{2}}$

Step 6

Factor $2$ out of $24 + 2 y$ .

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Step 6.1

Factor $2$ out of $24$ .

$\frac{2 \cdot 12 + 2 y}{9 y} - \frac{3}{10 y^{2}}$

Step 6.2

Factor $2$ out of $2 \cdot 12 + 2 y$ .

$\frac{2 (12 + y)}{9 y} - \frac{3}{10 y^{2}}$

Step 7

To write $\frac{2 (12 + y)}{9 y}$ as a fraction with a common denominator, multiply by $\frac{10 y}{10 y}$ .

$\frac{2 (12 + y)}{9 y} \cdot \frac{10 y}{10 y} - \frac{3}{10 y^{2}}$

Step 8

To write $- \frac{3}{10 y^{2}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 (12 + y)}{9 y} \cdot \frac{10 y}{10 y} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9

Write each expression with a common denominator of $90 y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{2 (12 + y)}{9 y}$ by $\frac{10 y}{10 y}$ .

$\frac{2 (12 + y) (10 y)}{9 y (10 y)} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.2

Multiply $10$ by $9$ .

$\frac{2 (12 + y) (10 y)}{90 y \cdot y} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.3

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

$\frac{2 (12 + y) (10 y)}{90 (y^{1} y)} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.4

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

$\frac{2 (12 + y) (10 y)}{90 (y^{1} y^{1})} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.5

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

$\frac{2 (12 + y) (10 y)}{90 y^{1 + 1}} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.6

Add $1$ and $1$ .

$\frac{2 (12 + y) (10 y)}{90 y^{2}} - \frac{3}{10 y^{2}} \cdot \frac{9}{9}$

Step 9.7

Multiply $\frac{3}{10 y^{2}}$ by $\frac{9}{9}$ .

$\frac{2 (12 + y) (10 y)}{90 y^{2}} - \frac{3 \cdot 9}{10 y^{2} \cdot 9}$

Step 9.8

Multiply $9$ by $10$ .

$\frac{2 (12 + y) (10 y)}{90 y^{2}} - \frac{3 \cdot 9}{90 y^{2}}$

Step 10

Combine the numerators over the common denominator.

$\frac{2 (12 + y) (10 y) - 3 \cdot 9}{90 y^{2}}$

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

$\frac{(2 \cdot 12 + 2 y) (10 y) - 3 \cdot 9}{90 y^{2}}$

Step 11.2

Multiply $2$ by $12$ .

$\frac{(24 + 2 y) (10 y) - 3 \cdot 9}{90 y^{2}}$

Step 11.3

Apply the distributive property.

$\frac{24 (10 y) + 2 y (10 y) - 3 \cdot 9}{90 y^{2}}$

Step 11.4

Multiply $10$ by $24$ .

$\frac{240 y + 2 y (10 y) - 3 \cdot 9}{90 y^{2}}$

Step 11.5

Rewrite using the commutative property of multiplication.

$\frac{240 y + 2 \cdot 10 y \cdot y - 3 \cdot 9}{90 y^{2}}$

Step 11.6

Simplify each term.

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Step 11.6.1

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

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Step 11.6.1.1

Move $y$ .

$\frac{240 y + 2 \cdot 10 (y \cdot y) - 3 \cdot 9}{90 y^{2}}$

Step 11.6.1.2

Multiply $y$ by $y$ .

$\frac{240 y + 2 \cdot 10 y^{2} - 3 \cdot 9}{90 y^{2}}$

Step 11.6.2

Multiply $2$ by $10$ .

$\frac{240 y + 20 y^{2} - 3 \cdot 9}{90 y^{2}}$

Step 11.7

Multiply $- 3$ by $9$ .

$\frac{240 y + 20 y^{2} - 27}{90 y^{2}}$

Step 11.8

Reorder terms.

$\frac{20 y^{2} + 240 y - 27}{90 y^{2}}$