Algebra Examples

$\frac{5 b}{6 a} + \frac{3 b}{10 a^{2}} + \frac{2}{a b^{2}}$

Step 1

To write $\frac{5 b}{6 a}$ as a fraction with a common denominator, multiply by $\frac{5 a}{5 a}$ .

$\frac{5 b}{6 a} \cdot \frac{5 a}{5 a} + \frac{3 b}{10 a^{2}} + \frac{2}{a b^{2}}$

Step 2

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

$\frac{5 b}{6 a} \cdot \frac{5 a}{5 a} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3

Write each expression with a common denominator of $30 a^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5 b}{6 a}$ by $\frac{5 a}{5 a}$ .

$\frac{5 b (5 a)}{6 a (5 a)} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.2

Multiply $5$ by $6$ .

$\frac{5 b (5 a)}{30 a \cdot a} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.3

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

$\frac{5 b (5 a)}{30 (a^{1} a)} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.4

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

$\frac{5 b (5 a)}{30 (a^{1} a^{1})} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.5

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

$\frac{5 b (5 a)}{30 a^{1 + 1}} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.6

Add $1$ and $1$ .

$\frac{5 b (5 a)}{30 a^{2}} + \frac{3 b}{10 a^{2}} \cdot \frac{3}{3} + \frac{2}{a b^{2}}$

Step 3.7

Multiply $\frac{3 b}{10 a^{2}}$ by $\frac{3}{3}$ .

$\frac{5 b (5 a)}{30 a^{2}} + \frac{3 b \cdot 3}{10 a^{2} \cdot 3} + \frac{2}{a b^{2}}$

Step 3.8

Multiply $3$ by $10$ .

$\frac{5 b (5 a)}{30 a^{2}} + \frac{3 b \cdot 3}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 b (5 a) + 3 b \cdot 3}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

$\frac{b (5 \cdot 5 a) + 3 b \cdot 3}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 5.1.2

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

$\frac{b (5 \cdot 5 a) + b (3 \cdot 3)}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 5.1.3

Factor $b$ out of $b (5 \cdot 5 a) + b (3 \cdot 3)$ .

$\frac{b (5 \cdot 5 a + 3 \cdot 3)}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 5.2

Multiply $5$ by $5$ .

$\frac{b (25 a + 3 \cdot 3)}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 5.3

Multiply $3$ by $3$ .

$\frac{b (25 a + 9)}{30 a^{2}} + \frac{2}{a b^{2}}$

Step 6

To write $\frac{b (25 a + 9)}{30 a^{2}}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

$\frac{b (25 a + 9)}{30 a^{2}} \cdot \frac{b^{2}}{b^{2}} + \frac{2}{a b^{2}}$

Step 7

To write $\frac{2}{a b^{2}}$ as a fraction with a common denominator, multiply by $\frac{30 a}{30 a}$ .

$\frac{b (25 a + 9)}{30 a^{2}} \cdot \frac{b^{2}}{b^{2}} + \frac{2}{a b^{2}} \cdot \frac{30 a}{30 a}$

Step 8

Write each expression with a common denominator of $30 a^{2} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{b (25 a + 9)}{30 a^{2}}$ by $\frac{b^{2}}{b^{2}}$ .

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2}{a b^{2}} \cdot \frac{30 a}{30 a}$

Step 8.2

Multiply $\frac{2}{a b^{2}}$ by $\frac{30 a}{30 a}$ .

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{a b^{2} (30 a)}$

Step 8.3

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

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{a^{1} a b^{2} \cdot 30}$

Step 8.4

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

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{a^{1} a^{1} b^{2} \cdot 30}$

Step 8.5

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

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{a^{1 + 1} b^{2} \cdot 30}$

Step 8.6

Add $1$ and $1$ .

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{a^{2} b^{2} \cdot 30}$

Step 8.7

Reorder the factors of $a^{2} b^{2} \cdot 30$ .

$\frac{b (25 a + 9) b^{2}}{30 a^{2} b^{2}} + \frac{2 (30 a)}{30 a^{2} b^{2}}$

Step 9

Combine the numerators over the common denominator.

$\frac{b (25 a + 9) b^{2} + 2 (30 a)}{30 a^{2} b^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

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

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Step 10.1.1

Move $b^{2}$ .

$\frac{b^{2} b (25 a + 9) + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.1.2

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

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Step 10.1.2.1

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

$\frac{b^{2} b^{1} (25 a + 9) + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.1.2.2

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

$\frac{b^{2 + 1} (25 a + 9) + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.1.3

Add $2$ and $1$ .

$\frac{b^{3} (25 a + 9) + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.2

Apply the distributive property.

$\frac{b^{3} (25 a) + b^{3} \cdot 9 + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.3

Rewrite using the commutative property of multiplication.

$\frac{25 b^{3} a + b^{3} \cdot 9 + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.4

Move $9$ to the left of $b^{3}$ .

$\frac{25 b^{3} a + 9 \cdot b^{3} + 2 \cdot 30 a}{30 a^{2} b^{2}}$

Step 10.5

Multiply $2$ by $30$ .

$\frac{25 b^{3} a + 9 b^{3} + 60 a}{30 a^{2} b^{2}}$