Precalculus Examples

$\frac{1}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$

Step 1

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

$\frac{1}{x} \cdot \frac{x}{x} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$

Step 2

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{1}{x}$ by $\frac{x}{x}$ .

$\frac{x}{x \cdot x} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$

Step 2.2

Multiply $x$ by $x$ .

$\frac{x}{x^{2}} + \frac{1}{x^{2}} + \frac{1}{x^{3}}$

Step 3

Combine the numerators over the common denominator.

$\frac{x + 1}{x^{2}} + \frac{1}{x^{3}}$

Step 4

To write $\frac{x + 1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x + 1}{x^{2}} \cdot \frac{x}{x} + \frac{1}{x^{3}}$

Step 5

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

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Step 5.1

Multiply $\frac{x + 1}{x^{2}}$ by $\frac{x}{x}$ .

$\frac{(x + 1) x}{x^{2} x} + \frac{1}{x^{3}}$

Step 5.2

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

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Step 5.2.1

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

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Step 5.2.1.1

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

$\frac{(x + 1) x}{x^{2} x^{1}} + \frac{1}{x^{3}}$

Step 5.2.1.2

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

$\frac{(x + 1) x}{x^{2 + 1}} + \frac{1}{x^{3}}$

Step 5.2.2

Add $2$ and $1$ .

$\frac{(x + 1) x}{x^{3}} + \frac{1}{x^{3}}$

Step 6

Combine the numerators over the common denominator.

$\frac{(x + 1) x + 1}{x^{3}}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{x \cdot x + 1 x + 1}{x^{3}}$

Step 7.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 1 x + 1}{x^{3}}$

Step 7.3

Multiply $x$ by $1$ .

$\frac{x^{2} + x + 1}{x^{3}}$