Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $a$ out of $5 a b$ .

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

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $a$ out of $3 a^{2} b^{3}$ .

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

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

Step 1.2

Cancel the common factor of $b$ and $b^{3}$ .

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Step 1.2.1

Factor $b$ out of $5 b$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $b$ out of $3 a b^{3}$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.3

Cancel the common factor of $a$ and $a^{4}$ .

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Step 1.3.1

Factor $a$ out of $5 a b$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

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

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Step 1.4.1

Factor $b$ out of $5 b$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $b$ out of $2 a^{3} b^{2}$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.5

Cancel the common factor of $a$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

Cancel the common factor of $b$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

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

Step 2

To write $\frac{5}{3 a b^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 a^{2}}{2 a^{2}}$ .

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

Step 3

To write $\frac{5}{2 a^{3} b}$ as a fraction with a common denominator, multiply by $\frac{3 b}{3 b}$ .

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

Step 4

Write each expression with a common denominator of $6 a^{3} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5}{3 a b^{2}}$ by $\frac{2 a^{2}}{2 a^{2}}$ .

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

Step 4.2

Multiply $2$ by $3$ .

$\frac{5 (2 a^{2})}{6 a b^{2} a^{2}} + \frac{5}{2 a^{3} b} \cdot \frac{3 b}{3 b} + \frac{5}{4}$

Step 4.3

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

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Step 4.3.1

Move $a^{2}$ .

$\frac{5 (2 a^{2})}{6 (a^{2} a) b^{2}} + \frac{5}{2 a^{3} b} \cdot \frac{3 b}{3 b} + \frac{5}{4}$

Step 4.3.2

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

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Step 4.3.2.1

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

$\frac{5 (2 a^{2})}{6 (a^{2} a^{1}) b^{2}} + \frac{5}{2 a^{3} b} \cdot \frac{3 b}{3 b} + \frac{5}{4}$

Step 4.3.2.2

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

$\frac{5 (2 a^{2})}{6 a^{2 + 1} b^{2}} + \frac{5}{2 a^{3} b} \cdot \frac{3 b}{3 b} + \frac{5}{4}$

Step 4.3.3

Add $2$ and $1$ .

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5}{2 a^{3} b} \cdot \frac{3 b}{3 b} + \frac{5}{4}$

Step 4.4

Multiply $\frac{5}{2 a^{3} b}$ by $\frac{3 b}{3 b}$ .

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{2 a^{3} b (3 b)} + \frac{5}{4}$

Step 4.5

Multiply $3$ by $2$ .

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{6 a^{3} b \cdot b} + \frac{5}{4}$

Step 4.6

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

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{6 a^{3} (b^{1} b)} + \frac{5}{4}$

Step 4.7

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

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{6 a^{3} (b^{1} b^{1})} + \frac{5}{4}$

Step 4.8

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

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{6 a^{3} b^{1 + 1}} + \frac{5}{4}$

Step 4.9

Add $1$ and $1$ .

$\frac{5 (2 a^{2})}{6 a^{3} b^{2}} + \frac{5 (3 b)}{6 a^{3} b^{2}} + \frac{5}{4}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (2 a^{2}) + 5 (3 b)}{6 a^{3} b^{2}} + \frac{5}{4}$

Step 6

Factor $5$ out of $5 \cdot 2 a^{2} + 5 \cdot 3 b$ .

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Step 6.1

Factor $5$ out of $5 \cdot 2 a^{2}$ .

$\frac{5 (2 a^{2}) + 5 \cdot 3 b}{6 a^{3} b^{2}} + \frac{5}{4}$

Step 6.2

Factor $5$ out of $5 \cdot 3 b$ .

$\frac{5 (2 a^{2}) + 5 (3 b)}{6 a^{3} b^{2}} + \frac{5}{4}$

Step 6.3

Factor $5$ out of $5 (2 a^{2}) + 5 (3 b)$ .

$\frac{5 (2 a^{2} + 3 b)}{6 a^{3} b^{2}} + \frac{5}{4}$

Step 7

To write $\frac{5 (2 a^{2} + 3 b)}{6 a^{3} b^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 (2 a^{2} + 3 b)}{6 a^{3} b^{2}} \cdot \frac{2}{2} + \frac{5}{4}$

Step 8

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{3 a^{3} b^{2}}{3 a^{3} b^{2}}$ .

$\frac{5 (2 a^{2} + 3 b)}{6 a^{3} b^{2}} \cdot \frac{2}{2} + \frac{5}{4} \cdot \frac{3 a^{3} b^{2}}{3 a^{3} b^{2}}$

Step 9

Write each expression with a common denominator of $12 a^{3} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{5 (2 a^{2} + 3 b)}{6 a^{3} b^{2}}$ by $\frac{2}{2}$ .

$\frac{5 (2 a^{2} + 3 b) \cdot 2}{6 a^{3} b^{2} \cdot 2} + \frac{5}{4} \cdot \frac{3 a^{3} b^{2}}{3 a^{3} b^{2}}$

Step 9.2

Multiply $2$ by $6$ .

$\frac{5 (2 a^{2} + 3 b) \cdot 2}{12 a^{3} b^{2}} + \frac{5}{4} \cdot \frac{3 a^{3} b^{2}}{3 a^{3} b^{2}}$

Step 9.3

Multiply $\frac{5}{4}$ by $\frac{3 a^{3} b^{2}}{3 a^{3} b^{2}}$ .

$\frac{5 (2 a^{2} + 3 b) \cdot 2}{12 a^{3} b^{2}} + \frac{5 (3 a^{3} b^{2})}{4 (3 a^{3} b^{2})}$

Step 9.4

Multiply $3$ by $4$ .

$\frac{5 (2 a^{2} + 3 b) \cdot 2}{12 a^{3} b^{2}} + \frac{5 (3 a^{3} b^{2})}{12 (a^{3} b^{2})}$

Step 9.5

Reorder the factors of $12 (a^{3} b^{2})$ .

$\frac{5 (2 a^{2} + 3 b) \cdot 2}{12 a^{3} b^{2}} + \frac{5 (3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 10

Combine the numerators over the common denominator.

$\frac{5 (2 a^{2} + 3 b) \cdot 2 + 5 (3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $5$ out of $5 (2 a^{2} + 3 b) \cdot 2 + 5 \cdot 3 a^{3} b^{2}$ .

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Step 11.1.1

Factor $5$ out of $5 (2 a^{2} + 3 b) \cdot 2$ .

$\frac{5 ((2 a^{2} + 3 b) \cdot 2) + 5 \cdot 3 a^{3} b^{2}}{12 a^{3} b^{2}}$

Step 11.1.2

Factor $5$ out of $5 \cdot 3 a^{3} b^{2}$ .

$\frac{5 ((2 a^{2} + 3 b) \cdot 2) + 5 (3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 11.1.3

Factor $5$ out of $5 ((2 a^{2} + 3 b) \cdot 2) + 5 (3 a^{3} b^{2})$ .

$\frac{5 ((2 a^{2} + 3 b) \cdot 2 + 3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 11.2

Apply the distributive property.

$\frac{5 (2 a^{2} \cdot 2 + 3 b \cdot 2 + 3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 11.3

Multiply $2$ by $2$ .

$\frac{5 (4 a^{2} + 3 b \cdot 2 + 3 a^{3} b^{2})}{12 a^{3} b^{2}}$

Step 11.4

Multiply $2$ by $3$ .

$\frac{5 (4 a^{2} + 6 b + 3 a^{3} b^{2})}{12 a^{3} b^{2}}$