Precalculus Examples

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

Step 1

Factor $x$ out of $x^{2} + x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $x$ out of $x^{1}$ .

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

Step 1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $\frac{1}{x (x + 1)}$ by $\frac{x}{x}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Add $x$ and $x$ .

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