Precalculus Examples

$\frac{\frac{2}{a} + 4}{4 + \frac{2}{b}}$

Step 1

Cancel the common factor of $\frac{2}{a} + 4$ and $4 + \frac{2}{b}$ .

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Step 1.1

Factor $2$ out of $\frac{2}{a}$ .

$\frac{2 \frac{1}{a} + 4}{4 + \frac{2}{b}}$

Step 1.2

Factor $2$ out of $4$ .

$\frac{2 \frac{1}{a} + 2 \cdot 2}{4 + \frac{2}{b}}$

Step 1.3

Factor $2$ out of $2 \frac{1}{a} + 2 \cdot 2$ .

$\frac{2 (\frac{1}{a} + 2)}{4 + \frac{2}{b}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $4$ .

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

Step 1.4.2

Factor $2$ out of $\frac{2}{b}$ .

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

Step 1.4.3

Factor $2$ out of $2 \cdot 2 + 2 \frac{1}{b}$ .

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

Step 1.4.4

Cancel the common factor.

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

Step 1.4.5

Rewrite the expression.

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

Step 2

Multiply the numerator and denominator of the fraction by $a b$ .

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Step 2.1

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

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

Step 2.2

Combine.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify by cancelling.

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Step 4.1

Cancel the common factor of $a$ .

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Step 4.1.1

Factor $a$ out of $a b$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $b$ .

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Step 4.2.1

Factor $b$ out of $a b$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $b$ out of $b + a b \cdot 2$ .

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Step 5.1.1

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

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

Step 5.1.2

Factor $b$ out of $b^{1}$ .

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

Step 5.1.3

Factor $b$ out of $a b \cdot 2$ .

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

Step 5.1.4

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

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

Step 5.2

Move $2$ to the left of $a$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Factor $a$ out of $a b \cdot 2 + a$ .

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Step 6.1.1

Factor $a$ out of $a b \cdot 2$ .

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

Step 6.1.2

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

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

Step 6.1.3

Factor $a$ out of $a^{1}$ .

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

Step 6.1.4

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

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

Step 6.2

Move $2$ to the left of $b$ .

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