Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

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

Tap for more steps...

Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.2

Add $2$ and $1$ .

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

Step 4.2

Rewrite $1$ as $1^{3}$ .

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

Step 4.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Multiply $- 1$ by $1$ .

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

Step 4.4.2

One to any power is one.

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

Step 5

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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