Algebra Examples

$\frac{5}{6 a b} + \frac{3 b^{2}}{14 a^{3}}$

Step 1

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

$\frac{5}{6 a b} \cdot \frac{7 a^{2}}{7 a^{2}} + \frac{3 b^{2}}{14 a^{3}}$

Step 2

To write $\frac{3 b^{2}}{14 a^{3}}$ as a fraction with a common denominator, multiply by $\frac{3 b}{3 b}$ .

$\frac{5}{6 a b} \cdot \frac{7 a^{2}}{7 a^{2}} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3

Write each expression with a common denominator of $42 b a^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{6 a b}$ by $\frac{7 a^{2}}{7 a^{2}}$ .

$\frac{5 (7 a^{2})}{6 a b (7 a^{2})} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.2

Multiply $7$ by $6$ .

$\frac{5 (7 a^{2})}{42 a b a^{2}} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.3

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

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Step 3.3.1

Move $a^{2}$ .

$\frac{5 (7 a^{2})}{42 (a^{2} a) b} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.3.2

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

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Step 3.3.2.1

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

$\frac{5 (7 a^{2})}{42 (a^{2} a^{1}) b} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.3.2.2

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

$\frac{5 (7 a^{2})}{42 a^{2 + 1} b} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.3.3

Add $2$ and $1$ .

$\frac{5 (7 a^{2})}{42 a^{3} b} + \frac{3 b^{2}}{14 a^{3}} \cdot \frac{3 b}{3 b}$

Step 3.4

Multiply $\frac{3 b^{2}}{14 a^{3}}$ by $\frac{3 b}{3 b}$ .

$\frac{5 (7 a^{2})}{42 a^{3} b} + \frac{3 b^{2} (3 b)}{14 a^{3} (3 b)}$

Step 3.5

Multiply $3$ by $14$ .

$\frac{5 (7 a^{2})}{42 a^{3} b} + \frac{3 b^{2} (3 b)}{42 a^{3} b}$

Step 4

Combine the numerators over the common denominator.

$\frac{5 (7 a^{2}) + 3 b^{2} (3 b)}{42 a^{3} b}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $5$ by $7$ .

$\frac{35 a^{2} + 3 b^{2} (3 b)}{42 a^{3} b}$

Step 5.2

Rewrite using the commutative property of multiplication.

$\frac{35 a^{2} + 3 \cdot 3 b^{2} b}{42 a^{3} b}$

Step 5.3

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

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Step 5.3.1

Move $b$ .

$\frac{35 a^{2} + 3 \cdot 3 (b \cdot b^{2})}{42 a^{3} b}$

Step 5.3.2

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

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Step 5.3.2.1

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

$\frac{35 a^{2} + 3 \cdot 3 (b^{1} b^{2})}{42 a^{3} b}$

Step 5.3.2.2

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

$\frac{35 a^{2} + 3 \cdot 3 b^{1 + 2}}{42 a^{3} b}$

Step 5.3.3

Add $1$ and $2$ .

$\frac{35 a^{2} + 3 \cdot 3 b^{3}}{42 a^{3} b}$

Step 5.4

Multiply $3$ by $3$ .

$\frac{35 a^{2} + 9 b^{3}}{42 a^{3} b}$