Algebra Examples

$\frac{3 b + 2 c}{9 b^{2} c} - \frac{2 c - 5 b}{6 b c^{2}}$

Step 1

To write $\frac{3 b + 2 c}{9 b^{2} c}$ as a fraction with a common denominator, multiply by $\frac{2 c}{2 c}$ .

$\frac{3 b + 2 c}{9 b^{2} c} \cdot \frac{2 c}{2 c} - \frac{2 c - 5 b}{6 b c^{2}}$

Step 2

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

$\frac{3 b + 2 c}{9 b^{2} c} \cdot \frac{2 c}{2 c} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3

Write each expression with a common denominator of $18 b^{2} c^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3 b + 2 c}{9 b^{2} c}$ by $\frac{2 c}{2 c}$ .

$\frac{(3 b + 2 c) (2 c)}{9 b^{2} c (2 c)} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.2

Multiply $2$ by $9$ .

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c \cdot c} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.3

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} (c^{1} c)} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.4

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} (c^{1} c^{1})} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.5

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{1 + 1}} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.6

Add $1$ and $1$ .

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{2 c - 5 b}{6 b c^{2}} \cdot \frac{3 b}{3 b}$

Step 3.7

Multiply $\frac{2 c - 5 b}{6 b c^{2}}$ by $\frac{3 b}{3 b}$ .

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{6 b c^{2} (3 b)}$

Step 3.8

Multiply $3$ by $6$ .

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{18 b c^{2} b}$

Step 3.9

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{18 (b^{1} b) c^{2}}$

Step 3.10

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{18 (b^{1} b^{1}) c^{2}}$

Step 3.11

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

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{18 b^{1 + 1} c^{2}}$

Step 3.12

Add $1$ and $1$ .

$\frac{(3 b + 2 c) (2 c)}{18 b^{2} c^{2}} - \frac{(2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(3 b + 2 c) (2 c) - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(3 b \cdot 2 + 2 c \cdot 2) c - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.2

Multiply $2$ by $3$ .

$\frac{(6 b + 2 c \cdot 2) c - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.3

Multiply $2$ by $2$ .

$\frac{(6 b + 4 c) c - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.4

Apply the distributive property.

$\frac{6 b c + 4 c \cdot c - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.5

Multiply $c$ by $c$ by adding the exponents.

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Step 5.5.1

Move $c$ .

$\frac{6 b c + 4 (c \cdot c) - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.5.2

Multiply $c$ by $c$ .

$\frac{6 b c + 4 c^{2} - (2 c - 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.6

Apply the distributive property.

$\frac{6 b c + 4 c^{2} + (- (2 c) - (- 5 b)) (3 b)}{18 b^{2} c^{2}}$

Step 5.7

Multiply $2$ by $- 1$ .

$\frac{6 b c + 4 c^{2} + (- 2 c - (- 5 b)) (3 b)}{18 b^{2} c^{2}}$

Step 5.8

Multiply $- 5$ by $- 1$ .

$\frac{6 b c + 4 c^{2} + (- 2 c + 5 b) (3 b)}{18 b^{2} c^{2}}$

Step 5.9

Apply the distributive property.

$\frac{6 b c + 4 c^{2} - 2 c (3 b) + 5 b (3 b)}{18 b^{2} c^{2}}$

Step 5.10

Rewrite using the commutative property of multiplication.

$\frac{6 b c + 4 c^{2} - 2 \cdot 3 c b + 5 b (3 b)}{18 b^{2} c^{2}}$

Step 5.11

Rewrite using the commutative property of multiplication.

$\frac{6 b c + 4 c^{2} - 2 \cdot 3 c b + 5 \cdot 3 b \cdot b}{18 b^{2} c^{2}}$

Step 5.12

Simplify each term.

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Step 5.12.1

Multiply $- 2$ by $3$ .

$\frac{6 b c + 4 c^{2} - 6 c b + 5 \cdot 3 b \cdot b}{18 b^{2} c^{2}}$

Step 5.12.2

Multiply $b$ by $b$ by adding the exponents.

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Step 5.12.2.1

Move $b$ .

$\frac{6 b c + 4 c^{2} - 6 c b + 5 \cdot 3 (b \cdot b)}{18 b^{2} c^{2}}$

Step 5.12.2.2

Multiply $b$ by $b$ .

$\frac{6 b c + 4 c^{2} - 6 c b + 5 \cdot 3 b^{2}}{18 b^{2} c^{2}}$

Step 5.12.3

Multiply $5$ by $3$ .

$\frac{6 b c + 4 c^{2} - 6 c b + 15 b^{2}}{18 b^{2} c^{2}}$

Step 5.13

Subtract $6 c b$ from $6 b c$ .

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Step 5.13.1

Move $c$ .

$\frac{4 c^{2} + 6 b c - 6 b c + 15 b^{2}}{18 b^{2} c^{2}}$

Step 5.13.2

Subtract $6 b c$ from $6 b c$ .

$\frac{4 c^{2} + 0 + 15 b^{2}}{18 b^{2} c^{2}}$

Step 5.14

Add $4 c^{2}$ and $0$ .

$\frac{4 c^{2} + 15 b^{2}}{18 b^{2} c^{2}}$