Algebra Examples

$\frac{7 x - 8}{9 x + 1} - \frac{1}{5 x^{3}}$

Step 1

To write $\frac{7 x - 8}{9 x + 1}$ as a fraction with a common denominator, multiply by $\frac{5 x^{3}}{5 x^{3}}$ .

$\frac{7 x - 8}{9 x + 1} \cdot \frac{5 x^{3}}{5 x^{3}} - \frac{1}{5 x^{3}}$

Step 2

To write $- \frac{1}{5 x^{3}}$ as a fraction with a common denominator, multiply by $\frac{9 x + 1}{9 x + 1}$ .

$\frac{7 x - 8}{9 x + 1} \cdot \frac{5 x^{3}}{5 x^{3}} - \frac{1}{5 x^{3}} \cdot \frac{9 x + 1}{9 x + 1}$

Step 3

Write each expression with a common denominator of $(9 x + 1) \cdot 5 x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{7 x - 8}{9 x + 1}$ by $\frac{5 x^{3}}{5 x^{3}}$ .

$\frac{(7 x - 8) (5 x^{3})}{(9 x + 1) (5 x^{3})} - \frac{1}{5 x^{3}} \cdot \frac{9 x + 1}{9 x + 1}$

Step 3.2

Multiply $\frac{1}{5 x^{3}}$ by $\frac{9 x + 1}{9 x + 1}$ .

$\frac{(7 x - 8) (5 x^{3})}{(9 x + 1) (5 x^{3})} - \frac{9 x + 1}{5 x^{3} (9 x + 1)}$

Step 3.3

Reorder the factors of $(9 x + 1) (5 x^{3})$ .

$\frac{(7 x - 8) (5 x^{3})}{5 x^{3} (9 x + 1)} - \frac{9 x + 1}{5 x^{3} (9 x + 1)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(7 x - 8) (5 x^{3}) - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{(7 x \cdot 5 - 8 \cdot 5) x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.2

Multiply $5$ by $7$ .

$\frac{(35 x - 8 \cdot 5) x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.3

Multiply $- 8$ by $5$ .

$\frac{(35 x - 40) x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.4

Apply the distributive property.

$\frac{35 x \cdot x^{3} - 40 x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.5

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 5.5.1

Move $x^{3}$ .

$\frac{35 (x^{3} x) - 40 x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.5.2

Multiply $x^{3}$ by $x$ .

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Step 5.5.2.1

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

$\frac{35 (x^{3} x^{1}) - 40 x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.5.2.2

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

$\frac{35 x^{3 + 1} - 40 x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.5.3

Add $3$ and $1$ .

$\frac{35 x^{4} - 40 x^{3} - (9 x + 1)}{5 x^{3} (9 x + 1)}$

Step 5.6

Apply the distributive property.

$\frac{35 x^{4} - 40 x^{3} - (9 x) - 1 \cdot 1}{5 x^{3} (9 x + 1)}$

Step 5.7

Multiply $9$ by $- 1$ .

$\frac{35 x^{4} - 40 x^{3} - 9 x - 1 \cdot 1}{5 x^{3} (9 x + 1)}$

Step 5.8

Multiply $- 1$ by $1$ .

$\frac{35 x^{4} - 40 x^{3} - 9 x - 1}{5 x^{3} (9 x + 1)}$