Algebra Examples

$x^{- 2} (x^{4} - 3 x^{3} + x^{- 1})$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}} (x^{4} - 3 x^{3} + x^{- 1})$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}} (x^{4} - 3 x^{3} + \frac{1}{x})$

Step 3

Multiply $\frac{1}{x^{2}}$ by $x^{4} - 3 x^{3} + \frac{1}{x}$ .

$\frac{x^{4} - 3 x^{3} + \frac{1}{x}}{x^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{- 3 x^{3} + \frac{x^{4} x}{x} + \frac{1}{x}}{x^{2}}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- 3 x^{3} + \frac{x^{4} x + 1}{x}}{x^{2}}$

Step 4.3

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

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Step 4.3.1

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

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Step 4.3.1.1

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

$\frac{- 3 x^{3} + \frac{x^{4} x^{1} + 1}{x}}{x^{2}}$

Step 4.3.1.2

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

$\frac{- 3 x^{3} + \frac{x^{4 + 1} + 1}{x}}{x^{2}}$

Step 4.3.2

Add $4$ and $1$ .

$\frac{- 3 x^{3} + \frac{x^{5} + 1}{x}}{x^{2}}$

Step 4.4

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

$\frac{- 3 x^{3} \cdot \frac{x}{x} + \frac{x^{5} + 1}{x}}{x^{2}}$

Step 4.5

Combine $- 3 x^{3}$ and $\frac{x}{x}$ .

$\frac{\frac{- 3 x^{3} x}{x} + \frac{x^{5} + 1}{x}}{x^{2}}$

Step 4.6

Combine the numerators over the common denominator.

$\frac{\frac{- 3 x^{3} x + x^{5} + 1}{x}}{x^{2}}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

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

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Step 4.7.1.1

Move $x$ .

$\frac{\frac{- 3 (x \cdot x^{3}) + x^{5} + 1}{x}}{x^{2}}$

Step 4.7.1.2

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

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Step 4.7.1.2.1

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

$\frac{\frac{- 3 (x^{1} x^{3}) + x^{5} + 1}{x}}{x^{2}}$

Step 4.7.1.2.2

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

$\frac{\frac{- 3 x^{1 + 3} + x^{5} + 1}{x}}{x^{2}}$

Step 4.7.1.3

Add $1$ and $3$ .

$\frac{\frac{- 3 x^{4} + x^{5} + 1}{x}}{x^{2}}$

Step 4.7.2

Reorder terms.

$\frac{\frac{x^{5} - 3 x^{4} + 1}{x}}{x^{2}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{5} - 3 x^{4} + 1}{x} \cdot \frac{1}{x^{2}}$

Step 6

Multiply $\frac{x^{5} - 3 x^{4} + 1}{x} \cdot \frac{1}{x^{2}}$ .

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Step 6.1

Multiply $\frac{x^{5} - 3 x^{4} + 1}{x}$ by $\frac{1}{x^{2}}$ .

$\frac{x^{5} - 3 x^{4} + 1}{x \cdot x^{2}}$

Step 6.2

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

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Step 6.2.1

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

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Step 6.2.1.1

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

$\frac{x^{5} - 3 x^{4} + 1}{x^{1} x^{2}}$

Step 6.2.1.2

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

$\frac{x^{5} - 3 x^{4} + 1}{x^{1 + 2}}$

Step 6.2.2

Add $1$ and $2$ .

$\frac{x^{5} - 3 x^{4} + 1}{x^{3}}$