Finite Math Examples

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

Step 1

Expand $(a + \frac{1}{b}) (b + \frac{1}{a})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $a$ .

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Step 2.1.1.1

Cancel the common factor.

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

Step 2.1.1.2

Rewrite the expression.

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

Step 2.1.2

Cancel the common factor of $b$ .

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Step 2.1.2.1

Cancel the common factor.

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

Step 2.1.2.2

Rewrite the expression.

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

Step 2.1.3

Multiply $\frac{1}{b}$ by $\frac{1}{a}$ .

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

Step 2.2

Add $1$ and $1$ .

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

Step 3

Reorder $b$ and $a$ .

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