Algebra Examples

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

Step 1

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

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

Step 2

Expand $(x^{2} + \frac{1}{x^{2}}) (x^{2} + \frac{1}{x^{2}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

Add $2$ and $2$ .

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

Step 3.1.2

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

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

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

Step 3.1.3

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

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Step 3.1.3.1

Cancel the common factor.

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

Step 3.1.3.2

Rewrite the expression.

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

Step 3.1.4

Combine.

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

Add $2$ and $2$ .

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

Step 3.1.6

Multiply $1$ by $1$ .

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

Step 3.2

Add $1$ and $1$ .

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