Basic Math Examples

$\frac{a + 3 b}{3 a b} + \frac{a^{2} b - 4 a b^{2}}{5 a^{2} b^{2}}$

Step 1

Simplify each term.

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Step 1.1

Factor $a b$ out of $a^{2} b - 4 a b^{2}$ .

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Step 1.1.1

Factor $a b$ out of $a^{2} b$ .

$\frac{a + 3 b}{3 a b} + \frac{a b (a) - 4 a b^{2}}{5 a^{2} b^{2}}$

Step 1.1.2

Factor $a b$ out of $- 4 a b^{2}$ .

$\frac{a + 3 b}{3 a b} + \frac{a b (a) + a b (- 4 b)}{5 a^{2} b^{2}}$

Step 1.1.3

Factor $a b$ out of $a b (a) + a b (- 4 b)$ .

$\frac{a + 3 b}{3 a b} + \frac{a b (a - 4 b)}{5 a^{2} b^{2}}$

Step 1.2

Cancel the common factor of $a$ and $a^{2}$ .

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Step 1.2.1

Factor $a$ out of $a b (a - 4 b)$ .

$\frac{a + 3 b}{3 a b} + \frac{a (b (a - 4 b))}{5 a^{2} b^{2}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $a$ out of $5 a^{2} b^{2}$ .

$\frac{a + 3 b}{3 a b} + \frac{a (b (a - 4 b))}{a (5 a b^{2})}$

Step 1.2.2.2

Cancel the common factor.

$\frac{a + 3 b}{3 a b} + \frac{a (b (a - 4 b))}{a (5 a b^{2})}$

Step 1.2.2.3

Rewrite the expression.

$\frac{a + 3 b}{3 a b} + \frac{b (a - 4 b)}{5 a b^{2}}$

Step 1.3

Cancel the common factors.

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Step 1.3.1

Factor $b$ out of $5 a b^{2}$ .

$\frac{a + 3 b}{3 a b} + \frac{b (a - 4 b)}{b (5 a b)}$

Step 1.3.2

Cancel the common factor.

$\frac{a + 3 b}{3 a b} + \frac{b (a - 4 b)}{b (5 a b)}$

Step 1.3.3

Rewrite the expression.

$\frac{a + 3 b}{3 a b} + \frac{a - 4 b}{5 a b}$

Step 2

To write $\frac{a + 3 b}{3 a b}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a + 3 b}{3 a b} \cdot \frac{5}{5} + \frac{a - 4 b}{5 a b}$

Step 3

To write $\frac{a - 4 b}{5 a b}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a + 3 b}{3 a b} \cdot \frac{5}{5} + \frac{a - 4 b}{5 a b} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $15 a b$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{a + 3 b}{3 a b}$ by $\frac{5}{5}$ .

$\frac{(a + 3 b) \cdot 5}{3 a b \cdot 5} + \frac{a - 4 b}{5 a b} \cdot \frac{3}{3}$

Step 4.2

Multiply $5$ by $3$ .

$\frac{(a + 3 b) \cdot 5}{15 a b} + \frac{a - 4 b}{5 a b} \cdot \frac{3}{3}$

Step 4.3

Multiply $\frac{a - 4 b}{5 a b}$ by $\frac{3}{3}$ .

$\frac{(a + 3 b) \cdot 5}{15 a b} + \frac{(a - 4 b) \cdot 3}{5 a b \cdot 3}$

Step 4.4

Multiply $3$ by $5$ .

$\frac{(a + 3 b) \cdot 5}{15 a b} + \frac{(a - 4 b) \cdot 3}{15 a b}$

Step 5

Combine the numerators over the common denominator.

$\frac{(a + 3 b) \cdot 5 + (a - 4 b) \cdot 3}{15 a b}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{a \cdot 5 + 3 b \cdot 5 + (a - 4 b) \cdot 3}{15 a b}$

Step 6.2

Move $5$ to the left of $a$ .

$\frac{5 \cdot a + 3 b \cdot 5 + (a - 4 b) \cdot 3}{15 a b}$

Step 6.3

Multiply $5$ by $3$ .

$\frac{5 \cdot a + 15 b + (a - 4 b) \cdot 3}{15 a b}$

Step 6.4

Apply the distributive property.

$\frac{5 a + 15 b + a \cdot 3 - 4 b \cdot 3}{15 a b}$

Step 6.5

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

$\frac{5 a + 15 b + 3 \cdot a - 4 b \cdot 3}{15 a b}$

Step 6.6

Multiply $3$ by $- 4$ .

$\frac{5 a + 15 b + 3 \cdot a - 12 b}{15 a b}$

Step 6.7

Add $5 a$ and $3 a$ .

$\frac{8 a + 15 b - 12 b}{15 a b}$

Step 6.8

Subtract $12 b$ from $15 b$ .

$\frac{8 a + 3 b}{15 a b}$