Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $x^{2}$ and $x^{4}$ .

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Step 1.1.1

Multiply by $1$ .

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

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

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

Step 1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $2$ by $2$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine $- x^{4}$ and $\frac{x^{2}}{x^{2}}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Rewrite $x^{4} x^{2}$ as ${(x^{2})}^{3}$ .

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

Step 4.3

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

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

Step 4.4

Simplify.

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Step 4.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.4.2

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

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

Step 4.4.3

Multiply $x^{2}$ by $1$ .

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

Step 4.5

Simplify each term.

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Step 4.5.1

One to any power is one.

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

Step 4.5.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.5.2.2

Multiply $2$ by $2$ .

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