Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $a \cdot 2 b$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $a$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $b$ by $b^{2}$ by adding the exponents.

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Step 3.2.1

Move $b^{2}$ .

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

Step 3.2.2

Multiply $b^{2}$ by $b$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

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

Step 3.2.3

Add $2$ and $1$ .

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a \cdot 2 b \frac{a}{2} + a \cdot 2 b (- \frac{a}{b})}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{2 (a b) \frac{a}{2} + a \cdot 2 b (- \frac{a}{b})}$

Step 3.3.2

Cancel the common factor.

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{2 (a b) \frac{a}{2} + a \cdot 2 b (- \frac{a}{b})}$

Step 3.3.3

Rewrite the expression.

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a b a + a \cdot 2 b (- \frac{a}{b})}$

Step 3.4

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{1} a b + a \cdot 2 b (- \frac{a}{b})}$

Step 3.5

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{1} a^{1} b + a \cdot 2 b (- \frac{a}{b})}$

Step 3.6

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{1 + 1} b + a \cdot 2 b (- \frac{a}{b})}$

Step 3.7

Add $1$ and $1$ .

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + a \cdot 2 b (- \frac{a}{b})}$

Step 3.8

Cancel the common factor of $b$ .

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Step 3.8.1

Move the leading negative in $- \frac{a}{b}$ into the numerator.

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + a \cdot 2 b \frac{- a}{b}}$

Step 3.8.2

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + b (a \cdot 2) \frac{- a}{b}}$

Step 3.8.3

Cancel the common factor.

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + b (a \cdot 2) \frac{- a}{b}}$

Step 3.8.4

Rewrite the expression.

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + a \cdot 2 (- a)}$

Step 3.9

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + 2 (- (a^{1} a))}$

Step 3.10

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + 2 (- (a^{1} a^{1}))}$

Step 3.11

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

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + 2 (- a^{1 + 1})}$

Step 3.12

Add $1$ and $1$ .

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b + 2 (- a^{2})}$

Step 3.13

Multiply $- 1$ by $2$ .

$\frac{a \cdot 2 b \cdot 1 + 2 b^{3}}{a^{2} b - 2 a^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $2 b$ out of $a \cdot 2 b \cdot 1 + 2 b^{3}$ .

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Step 4.1.1

Factor $2 b$ out of $a \cdot 2 b \cdot 1$ .

$\frac{2 b (a \cdot 1) + 2 b^{3}}{a^{2} b - 2 a^{2}}$

Step 4.1.2

Factor $2 b$ out of $2 b^{3}$ .

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

Step 4.1.3

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

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

Step 4.2

Multiply $a$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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