Algebra Examples

$\frac{\frac{1}{b^{2}} + a^{2}}{\frac{1}{a^{2}} + b^{2}}$

Step 1

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

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Step 1.1

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

$\frac{b^{2} a^{2}}{b^{2} a^{2}} \cdot \frac{\frac{1}{b^{2}} + a^{2}}{\frac{1}{a^{2}} + b^{2}}$

Step 1.2

Combine.

$\frac{b^{2} a^{2} (\frac{1}{b^{2}} + a^{2})}{b^{2} a^{2} (\frac{1}{a^{2}} + b^{2})}$

Step 2

Apply the distributive property.

$\frac{b^{2} a^{2} \frac{1}{b^{2}} + b^{2} a^{2} a^{2}}{b^{2} a^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3

Simplify by cancelling.

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Step 3.1

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

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Step 3.1.1

Factor $b^{2}$ out of $b^{2} a^{2}$ .

$\frac{b^{2} (a^{2}) \frac{1}{b^{2}} + b^{2} a^{2} a^{2}}{b^{2} a^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3.1.2

Cancel the common factor.

$\frac{b^{2} a^{2} \frac{1}{b^{2}} + b^{2} a^{2} a^{2}}{b^{2} a^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3.1.3

Rewrite the expression.

$\frac{a^{2} + b^{2} a^{2} a^{2}}{b^{2} a^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3.2

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

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Step 3.2.1

Factor $a^{2}$ out of $b^{2} a^{2}$ .

$\frac{a^{2} + b^{2} a^{2} a^{2}}{a^{2} b^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3.2.2

Cancel the common factor.

$\frac{a^{2} + b^{2} a^{2} a^{2}}{a^{2} b^{2} \frac{1}{a^{2}} + b^{2} a^{2} b^{2}}$

Step 3.2.3

Rewrite the expression.

$\frac{a^{2} + b^{2} a^{2} a^{2}}{b^{2} + b^{2} a^{2} b^{2}}$

Step 4

Simplify terms.

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Step 4.1

Factor $a^{2}$ out of $a^{2} + b^{2} a^{2} a^{2}$ .

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Step 4.1.1

Multiply by $1$ .

$\frac{a^{2} \cdot 1 + b^{2} a^{2} a^{2}}{b^{2} + b^{2} a^{2} b^{2}}$

Step 4.1.2

Factor $a^{2}$ out of $b^{2} a^{2} a^{2}$ .

$\frac{a^{2} \cdot 1 + a^{2} (b^{2} a^{2})}{b^{2} + b^{2} a^{2} b^{2}}$

Step 4.1.3

Factor $a^{2}$ out of $a^{2} \cdot 1 + a^{2} (b^{2} a^{2})$ .

$\frac{a^{2} (1 + b^{2} a^{2})}{b^{2} + b^{2} a^{2} b^{2}}$

Step 4.2

Factor $b^{2}$ out of $b^{2} + b^{2} a^{2} b^{2}$ .

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Step 4.2.1

Multiply by $1$ .

$\frac{a^{2} (1 + b^{2} a^{2})}{b^{2} \cdot 1 + b^{2} a^{2} b^{2}}$

Step 4.2.2

Factor $b^{2}$ out of $b^{2} a^{2} b^{2}$ .

$\frac{a^{2} (1 + b^{2} a^{2})}{b^{2} \cdot 1 + b^{2} (a^{2} b^{2})}$

Step 4.2.3

Factor $b^{2}$ out of $b^{2} \cdot 1 + b^{2} (a^{2} b^{2})$ .

$\frac{a^{2} (1 + b^{2} a^{2})}{b^{2} (1 + a^{2} b^{2})}$

Step 4.3

Reorder the terms.

$\frac{a^{2} (a^{2} b^{2} + 1)}{b^{2} (a^{2} b^{2} + 1)}$

Step 4.4

Cancel the common factor of $a^{2} b^{2} + 1$ .

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Step 4.4.1

Cancel the common factor.

$\frac{a^{2} (a^{2} b^{2} + 1)}{b^{2} (a^{2} b^{2} + 1)}$

Step 4.4.2

Rewrite the expression.

$\frac{a^{2}}{b^{2}}$