Algebra Examples

$\frac{7 z^{2}}{12 z^{3} - 4 z^{2}} + \frac{3 z}{10 z + 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $4 z^{2}$ out of $12 z^{3} - 4 z^{2}$ .

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Step 1.1.1

Factor $4 z^{2}$ out of $12 z^{3}$ .

$\frac{7 z^{2}}{4 z^{2} (3 z) - 4 z^{2}} + \frac{3 z}{10 z + 25}$

Step 1.1.2

Factor $4 z^{2}$ out of $- 4 z^{2}$ .

$\frac{7 z^{2}}{4 z^{2} (3 z) + 4 z^{2} (- 1)} + \frac{3 z}{10 z + 25}$

Step 1.1.3

Factor $4 z^{2}$ out of $4 z^{2} (3 z) + 4 z^{2} (- 1)$ .

$\frac{7 z^{2}}{4 z^{2} (3 z - 1)} + \frac{3 z}{10 z + 25}$

Step 1.2

Cancel the common factor of $z^{2}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{7 z^{2}}{4 z^{2} (3 z - 1)} + \frac{3 z}{10 z + 25}$

Step 1.2.2

Rewrite the expression.

$\frac{7}{4 (3 z - 1)} + \frac{3 z}{10 z + 25}$

Step 1.3

Factor $5$ out of $10 z + 25$ .

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Step 1.3.1

Factor $5$ out of $10 z$ .

$\frac{7}{4 (3 z - 1)} + \frac{3 z}{5 (2 z) + 25}$

Step 1.3.2

Factor $5$ out of $25$ .

$\frac{7}{4 (3 z - 1)} + \frac{3 z}{5 (2 z) + 5 (5)}$

Step 1.3.3

Factor $5$ out of $5 (2 z) + 5 (5)$ .

$\frac{7}{4 (3 z - 1)} + \frac{3 z}{5 (2 z + 5)}$

Step 2

To write $\frac{7}{4 (3 z - 1)}$ as a fraction with a common denominator, multiply by $\frac{5 (2 z + 5)}{5 (2 z + 5)}$ .

$\frac{7}{4 (3 z - 1)} \cdot \frac{5 (2 z + 5)}{5 (2 z + 5)} + \frac{3 z}{5 (2 z + 5)}$

Step 3

To write $\frac{3 z}{5 (2 z + 5)}$ as a fraction with a common denominator, multiply by $\frac{4 (3 z - 1)}{4 (3 z - 1)}$ .

$\frac{7}{4 (3 z - 1)} \cdot \frac{5 (2 z + 5)}{5 (2 z + 5)} + \frac{3 z}{5 (2 z + 5)} \cdot \frac{4 (3 z - 1)}{4 (3 z - 1)}$

Step 4

Write each expression with a common denominator of $20 (3 z - 1) (2 z + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{7}{4 (3 z - 1)}$ by $\frac{5 (2 z + 5)}{5 (2 z + 5)}$ .

$\frac{7 (5 (2 z + 5))}{4 (3 z - 1) (5 (2 z + 5))} + \frac{3 z}{5 (2 z + 5)} \cdot \frac{4 (3 z - 1)}{4 (3 z - 1)}$

Step 4.2

Multiply $5$ by $4$ .

$\frac{7 (5 (2 z + 5))}{20 (3 z - 1) (2 z + 5)} + \frac{3 z}{5 (2 z + 5)} \cdot \frac{4 (3 z - 1)}{4 (3 z - 1)}$

Step 4.3

Multiply $\frac{3 z}{5 (2 z + 5)}$ by $\frac{4 (3 z - 1)}{4 (3 z - 1)}$ .

$\frac{7 (5 (2 z + 5))}{20 (3 z - 1) (2 z + 5)} + \frac{3 z (4 (3 z - 1))}{5 (2 z + 5) (4 (3 z - 1))}$

Step 4.4

Multiply $4$ by $5$ .

$\frac{7 (5 (2 z + 5))}{20 (3 z - 1) (2 z + 5)} + \frac{3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 4.5

Reorder the factors of $20 (3 z - 1) (2 z + 5)$ .

$\frac{7 (5 (2 z + 5))}{20 (2 z + 5) (3 z - 1)} + \frac{3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{7 (5 (2 z + 5)) + 3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $7$ by $5$ .

$\frac{35 (2 z + 5) + 3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 6.2

Apply the distributive property.

$\frac{35 (2 z) + 35 \cdot 5 + 3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 6.3

Multiply $2$ by $35$ .

$\frac{70 z + 35 \cdot 5 + 3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 6.4

Multiply $35$ by $5$ .

$\frac{70 z + 175 + 3 z (4 (3 z - 1))}{20 (2 z + 5) (3 z - 1)}$

Step 6.5

Apply the distributive property.

$\frac{70 z + 175 + 3 z (4 (3 z) + 4 \cdot - 1)}{20 (2 z + 5) (3 z - 1)}$

Step 6.6

Multiply $3$ by $4$ .

$\frac{70 z + 175 + 3 z (12 z + 4 \cdot - 1)}{20 (2 z + 5) (3 z - 1)}$

Step 6.7

Multiply $4$ by $- 1$ .

$\frac{70 z + 175 + 3 z (12 z - 4)}{20 (2 z + 5) (3 z - 1)}$

Step 6.8

Apply the distributive property.

$\frac{70 z + 175 + 3 z (12 z) + 3 z \cdot - 4}{20 (2 z + 5) (3 z - 1)}$

Step 6.9

Rewrite using the commutative property of multiplication.

$\frac{70 z + 175 + 3 \cdot 12 z \cdot z + 3 z \cdot - 4}{20 (2 z + 5) (3 z - 1)}$

Step 6.10

Multiply $- 4$ by $3$ .

$\frac{70 z + 175 + 3 \cdot 12 z \cdot z - 12 z}{20 (2 z + 5) (3 z - 1)}$

Step 6.11

Simplify each term.

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Step 6.11.1

Multiply $z$ by $z$ by adding the exponents.

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Step 6.11.1.1

Move $z$ .

$\frac{70 z + 175 + 3 \cdot 12 (z \cdot z) - 12 z}{20 (2 z + 5) (3 z - 1)}$

Step 6.11.1.2

Multiply $z$ by $z$ .

$\frac{70 z + 175 + 3 \cdot 12 z^{2} - 12 z}{20 (2 z + 5) (3 z - 1)}$

Step 6.11.2

Multiply $3$ by $12$ .

$\frac{70 z + 175 + 36 z^{2} - 12 z}{20 (2 z + 5) (3 z - 1)}$

Step 6.12

Subtract $12 z$ from $70 z$ .

$\frac{58 z + 175 + 36 z^{2}}{20 (2 z + 5) (3 z - 1)}$

Step 6.13

Reorder terms.

$\frac{36 z^{2} + 58 z + 175}{20 (2 z + 5) (3 z - 1)}$