Calculus Examples

$\frac{x^{2}}{x^{2 - 4}} - \frac{1}{x - 2}$

Step 1

Simplify each term.

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Step 1.1

Move $x^{2 - 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$x^{2} x^{- (2 - 4)} - \frac{1}{x - 2}$

Step 1.2

Multiply $x^{2}$ by $x^{- (2 - 4)}$ by adding the exponents.

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Step 1.2.1

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

$x^{2 - (2 - 4)} - \frac{1}{x - 2}$

Step 1.2.2

Subtract $4$ from $2$ .

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

Step 1.2.3

Multiply $- 1$ by $- 2$ .

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

Step 1.2.4

Add $2$ and $2$ .

$x^{4} - \frac{1}{x - 2}$

Step 2

To write $x^{4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x^{4} (x - 2)}{x - 2} - \frac{1}{x - 2}$

Step 3

Combine the numerators over the common denominator.

$\frac{x^{4} (x - 2) - 1}{x - 2}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x^{4} x + x^{4} \cdot - 2 - 1}{x - 2}$

Step 4.2

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 4.2.1

Multiply $x^{4}$ by $x$ .

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Step 4.2.1.1

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

$\frac{x^{4} x^{1} + x^{4} \cdot - 2 - 1}{x - 2}$

Step 4.2.1.2

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

$\frac{x^{4 + 1} + x^{4} \cdot - 2 - 1}{x - 2}$

Step 4.2.2

Add $4$ and $1$ .

$\frac{x^{5} + x^{4} \cdot - 2 - 1}{x - 2}$

Step 4.3

Move $- 2$ to the left of $x^{4}$ .

$\frac{x^{5} - 2 x^{4} - 1}{x - 2}$