Calculus Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

To write $\frac{1}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 3.2

To write $- \frac{1}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Step 3.3

Write each expression with a common denominator of $b a$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

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

Step 3.3.2

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

$\frac{\frac{a}{b a} - \frac{b}{a b}}{b}$

Step 3.3.3

Reorder the factors of $b a$ .

$\frac{\frac{a}{a b} - \frac{b}{a b}}{b}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{\frac{a - b}{a b}}{b}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{a - b}{a b} \cdot \frac{1}{b}$ .

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Step 5.1

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

$\frac{a - b}{a b \cdot b}$

Step 5.2

Raise $b$ to the power of $1$ .

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

Step 5.3

Raise $b$ to the power of $1$ .

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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