Algebra Examples

$\frac{6 z + 1}{4 z + 4} - \frac{2}{7 z^{2}}$

Step 1

Factor $4$ out of $4 z + 4$ .

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Step 1.1

Factor $4$ out of $4 z$ .

$\frac{6 z + 1}{4 (z) + 4} - \frac{2}{7 z^{2}}$

Step 1.2

Factor $4$ out of $4$ .

$\frac{6 z + 1}{4 (z) + 4 (1)} - \frac{2}{7 z^{2}}$

Step 1.3

Factor $4$ out of $4 (z) + 4 (1)$ .

$\frac{6 z + 1}{4 (z + 1)} - \frac{2}{7 z^{2}}$

Step 2

To write $\frac{6 z + 1}{4 (z + 1)}$ as a fraction with a common denominator, multiply by $\frac{7 z^{2}}{7 z^{2}}$ .

$\frac{6 z + 1}{4 (z + 1)} \cdot \frac{7 z^{2}}{7 z^{2}} - \frac{2}{7 z^{2}}$

Step 3

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

$\frac{6 z + 1}{4 (z + 1)} \cdot \frac{7 z^{2}}{7 z^{2}} - \frac{2}{7 z^{2}} \cdot \frac{4 (z + 1)}{4 (z + 1)}$

Step 4

Write each expression with a common denominator of $28 (z + 1) z^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6 z + 1}{4 (z + 1)}$ by $\frac{7 z^{2}}{7 z^{2}}$ .

$\frac{(6 z + 1) (7 z^{2})}{4 (z + 1) (7 z^{2})} - \frac{2}{7 z^{2}} \cdot \frac{4 (z + 1)}{4 (z + 1)}$

Step 4.2

Multiply $7$ by $4$ .

$\frac{(6 z + 1) (7 z^{2})}{28 (z + 1) z^{2}} - \frac{2}{7 z^{2}} \cdot \frac{4 (z + 1)}{4 (z + 1)}$

Step 4.3

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

$\frac{(6 z + 1) (7 z^{2})}{28 (z + 1) z^{2}} - \frac{2 (4 (z + 1))}{7 z^{2} (4 (z + 1))}$

Step 4.4

Multiply $4$ by $7$ .

$\frac{(6 z + 1) (7 z^{2})}{28 (z + 1) z^{2}} - \frac{2 (4 (z + 1))}{28 z^{2} (z + 1)}$

Step 4.5

Reorder the factors of $28 (z + 1) z^{2}$ .

$\frac{(6 z + 1) (7 z^{2})}{28 z^{2} (z + 1)} - \frac{2 (4 (z + 1))}{28 z^{2} (z + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(6 z + 1) (7 z^{2}) - 2 (4 (z + 1))}{28 z^{2} (z + 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{(6 z \cdot 7 + 1 \cdot 7) z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.2

Multiply $7$ by $6$ .

$\frac{(42 z + 1 \cdot 7) z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.3

Multiply $7$ by $1$ .

$\frac{(42 z + 7) z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.4

Apply the distributive property.

$\frac{42 z \cdot z^{2} + 7 z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.5

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

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Step 6.5.1

Move $z^{2}$ .

$\frac{42 (z^{2} z) + 7 z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.5.2

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

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Step 6.5.2.1

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

$\frac{42 (z^{2} z^{1}) + 7 z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.5.2.2

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

$\frac{42 z^{2 + 1} + 7 z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.5.3

Add $2$ and $1$ .

$\frac{42 z^{3} + 7 z^{2} - 2 \cdot 4 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.6

Multiply $- 2$ by $4$ .

$\frac{42 z^{3} + 7 z^{2} - 8 (z + 1)}{28 z^{2} (z + 1)}$

Step 6.7

Apply the distributive property.

$\frac{42 z^{3} + 7 z^{2} - 8 z - 8 \cdot 1}{28 z^{2} (z + 1)}$

Step 6.8

Multiply $- 8$ by $1$ .

$\frac{42 z^{3} + 7 z^{2} - 8 z - 8}{28 z^{2} (z + 1)}$