Algebra Examples

$\frac{x^{3} - 2 x}{14 x^{2}} \div \frac{5 x^{3} - 10 x}{- 7 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3} - 2 x}{14 x^{2}} \cdot \frac{- 7 x}{5 x^{3} - 10 x}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $7 x$ .

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Step 2.1.1

Factor $7 x$ out of $14 x^{2}$ .

$\frac{x^{3} - 2 x}{7 x (2 x)} \cdot \frac{- 7 x}{5 x^{3} - 10 x}$

Step 2.1.2

Factor $7 x$ out of $- 7 x$ .

$\frac{x^{3} - 2 x}{7 x (2 x)} \cdot \frac{7 x (- 1)}{5 x^{3} - 10 x}$

Step 2.1.3

Cancel the common factor.

$\frac{x^{3} - 2 x}{7 x (2 x)} \cdot \frac{7 x \cdot - 1}{5 x^{3} - 10 x}$

Step 2.1.4

Rewrite the expression.

$\frac{x^{3} - 2 x}{2 x} \cdot \frac{- 1}{5 x^{3} - 10 x}$

Step 2.2

Multiply $\frac{x^{3} - 2 x}{2 x}$ by $\frac{- 1}{5 x^{3} - 10 x}$ .

$\frac{(x^{3} - 2 x) \cdot - 1}{2 x (5 x^{3} - 10 x)}$

Step 2.3

Factor $x$ out of $x^{3} - 2 x$ .

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Step 2.3.1

Factor $x$ out of $x^{3}$ .

$\frac{(x \cdot x^{2} - 2 x) \cdot - 1}{2 x (5 x^{3} - 10 x)}$

Step 2.3.2

Factor $x$ out of $- 2 x$ .

$\frac{(x \cdot x^{2} + x \cdot - 2) \cdot - 1}{2 x (5 x^{3} - 10 x)}$

Step 2.3.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 2$ .

$\frac{x (x^{2} - 2) \cdot - 1}{2 x (5 x^{3} - 10 x)}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $5 x$ out of $5 x^{3} - 10 x$ .

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Step 3.1.1

Factor $5 x$ out of $5 x^{3}$ .

$\frac{x (x^{2} - 2) \cdot - 1}{2 x (5 x (x^{2}) - 10 x)}$

Step 3.1.2

Factor $5 x$ out of $- 10 x$ .

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

Step 3.1.3

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

$\frac{x (x^{2} - 2) \cdot - 1}{2 x (5 x (x^{2} - 2))}$

$\frac{x (x^{2} - 2) \cdot - 1}{2 x \cdot 5 x (x^{2} - 2)}$

Step 3.2

Combine exponents.

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Step 3.2.1

Multiply $5$ by $2$ .

$\frac{x (x^{2} - 2) \cdot - 1}{10 x \cdot x (x^{2} - 2)}$

Step 3.2.2

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

$\frac{x (x^{2} - 2) \cdot - 1}{10 (x^{1} x) (x^{2} - 2)}$

Step 3.2.3

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

$\frac{x (x^{2} - 2) \cdot - 1}{10 (x^{1} x^{1}) (x^{2} - 2)}$

Step 3.2.4

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

$\frac{x (x^{2} - 2) \cdot - 1}{10 x^{1 + 1} (x^{2} - 2)}$

Step 3.2.5

Add $1$ and $1$ .

$\frac{x (x^{2} - 2) \cdot - 1}{10 x^{2} (x^{2} - 2)}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.1.1

Factor $x$ out of $x (x^{2} - 2) \cdot - 1$ .

$\frac{x ((x^{2} - 2) \cdot - 1)}{10 x^{2} (x^{2} - 2)}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $x$ out of $10 x^{2} (x^{2} - 2)$ .

$\frac{x ((x^{2} - 2) \cdot - 1)}{x (10 x (x^{2} - 2))}$

Step 4.1.2.2

Cancel the common factor.

$\frac{x ((x^{2} - 2) \cdot - 1)}{x (10 x (x^{2} - 2))}$

Step 4.1.2.3

Rewrite the expression.

$\frac{(x^{2} - 2) \cdot - 1}{10 x (x^{2} - 2)}$

Step 4.2

Cancel the common factor of $x^{2} - 2$ .

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Step 4.2.1

Cancel the common factor.

$\frac{(x^{2} - 2) \cdot - 1}{10 x (x^{2} - 2)}$

Step 4.2.2

Rewrite the expression.

$\frac{- 1}{10 x}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{1}{10 x}$