Algebra Examples

$\frac{8 z^{4}}{8 z + 20} - \frac{2}{z^{5}}$

Step 1

Cancel the common factor of $8$ and $8 z + 20$ .

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Step 1.1

Factor $4$ out of $8 z^{4}$ .

$\frac{4 (2 z^{4})}{8 z + 20} - \frac{2}{z^{5}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $4$ out of $8 z$ .

$\frac{4 (2 z^{4})}{4 (2 z) + 20} - \frac{2}{z^{5}}$

Step 1.2.2

Factor $4$ out of $20$ .

$\frac{4 (2 z^{4})}{4 (2 z) + 4 (5)} - \frac{2}{z^{5}}$

Step 1.2.3

Factor $4$ out of $4 (2 z) + 4 (5)$ .

$\frac{4 (2 z^{4})}{4 (2 z + 5)} - \frac{2}{z^{5}}$

Step 1.2.4

Cancel the common factor.

$\frac{4 (2 z^{4})}{4 (2 z + 5)} - \frac{2}{z^{5}}$

Step 1.2.5

Rewrite the expression.

$\frac{2 z^{4}}{2 z + 5} - \frac{2}{z^{5}}$

Step 2

To write $\frac{2 z^{4}}{2 z + 5}$ as a fraction with a common denominator, multiply by $\frac{z^{5}}{z^{5}}$ .

$\frac{2 z^{4}}{2 z + 5} \cdot \frac{z^{5}}{z^{5}} - \frac{2}{z^{5}}$

Step 3

To write $- \frac{2}{z^{5}}$ as a fraction with a common denominator, multiply by $\frac{2 z + 5}{2 z + 5}$ .

$\frac{2 z^{4}}{2 z + 5} \cdot \frac{z^{5}}{z^{5}} - \frac{2}{z^{5}} \cdot \frac{2 z + 5}{2 z + 5}$

Step 4

Write each expression with a common denominator of $(2 z + 5) z^{5}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 z^{4}}{2 z + 5}$ by $\frac{z^{5}}{z^{5}}$ .

$\frac{2 z^{4} z^{5}}{(2 z + 5) z^{5}} - \frac{2}{z^{5}} \cdot \frac{2 z + 5}{2 z + 5}$

Step 4.2

Multiply $\frac{2}{z^{5}}$ by $\frac{2 z + 5}{2 z + 5}$ .

$\frac{2 z^{4} z^{5}}{(2 z + 5) z^{5}} - \frac{2 (2 z + 5)}{z^{5} (2 z + 5)}$

Step 4.3

Reorder the factors of $(2 z + 5) z^{5}$ .

$\frac{2 z^{4} z^{5}}{z^{5} (2 z + 5)} - \frac{2 (2 z + 5)}{z^{5} (2 z + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 z^{4} z^{5} - 2 (2 z + 5)}{z^{5} (2 z + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $2$ out of $2 z^{4} z^{5} - 2 (2 z + 5)$ .

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Step 6.1.1

Factor $2$ out of $2 z^{4} z^{5}$ .

$\frac{2 (z^{4} z^{5}) - 2 (2 z + 5)}{z^{5} (2 z + 5)}$

Step 6.1.2

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

$\frac{2 (z^{4} z^{5}) + 2 (- (2 z + 5))}{z^{5} (2 z + 5)}$

Step 6.1.3

Factor $2$ out of $2 (z^{4} z^{5}) + 2 (- (2 z + 5))$ .

$\frac{2 (z^{4} z^{5} - (2 z + 5))}{z^{5} (2 z + 5)}$

Step 6.2

Multiply $z^{4}$ by $z^{5}$ by adding the exponents.

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Step 6.2.1

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

$\frac{2 (z^{4 + 5} - (2 z + 5))}{z^{5} (2 z + 5)}$

Step 6.2.2

Add $4$ and $5$ .

$\frac{2 (z^{9} - (2 z + 5))}{z^{5} (2 z + 5)}$

Step 6.3

Apply the distributive property.

$\frac{2 (z^{9} - (2 z) - 1 \cdot 5)}{z^{5} (2 z + 5)}$

Step 6.4

Multiply $2$ by $- 1$ .

$\frac{2 (z^{9} - 2 z - 1 \cdot 5)}{z^{5} (2 z + 5)}$

Step 6.5

Multiply $- 1$ by $5$ .

$\frac{2 (z^{9} - 2 z - 5)}{z^{5} (2 z + 5)}$