Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Combine $a$ and $\frac{a b + 1}{b}$ .

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

Multiply $\frac{a (a b + 1)}{b}$ by $\frac{1}{b a + 1}$ .

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

Step 9

Cancel the common factor of $a b + 1$ and $b a + 1$ .

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Step 9.1

Reorder terms.

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$\frac{a}{b}$