Algebra Examples

$\frac{1}{x^{3}} - \frac{x}{y^{3}}$

Step 1

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

$\frac{1}{x^{3}} \cdot \frac{y^{3}}{y^{3}} - \frac{x}{y^{3}}$

Step 2

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

$\frac{1}{x^{3}} \cdot \frac{y^{3}}{y^{3}} - \frac{x}{y^{3}} \cdot \frac{x^{3}}{x^{3}}$

Step 3

Write each expression with a common denominator of $x^{3} y^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x^{3}}$ by $\frac{y^{3}}{y^{3}}$ .

$\frac{y^{3}}{x^{3} y^{3}} - \frac{x}{y^{3}} \cdot \frac{x^{3}}{x^{3}}$

Step 3.2

Multiply $\frac{x}{y^{3}}$ by $\frac{x^{3}}{x^{3}}$ .

$\frac{y^{3}}{x^{3} y^{3}} - \frac{x \cdot x^{3}}{y^{3} x^{3}}$

Step 3.3

Reorder the factors of $y^{3} x^{3}$ .

$\frac{y^{3}}{x^{3} y^{3}} - \frac{x \cdot x^{3}}{x^{3} y^{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{y^{3} - x \cdot x^{3}}{x^{3} y^{3}}$

Step 5

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

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Step 5.1

Move $x^{3}$ .

$\frac{y^{3} - (x^{3} x)}{x^{3} y^{3}}$

Step 5.2

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

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Step 5.2.1

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

$\frac{y^{3} - (x^{3} x^{1})}{x^{3} y^{3}}$

Step 5.2.2

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

$\frac{y^{3} - x^{3 + 1}}{x^{3} y^{3}}$

Step 5.3

Add $3$ and $1$ .

$\frac{y^{3} - x^{4}}{x^{3} y^{3}}$