Simplify (1+(b^2)/a)/(a/2-a/b)

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

Step 1

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

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

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

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.

Cancel the common factor of $a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Move $b^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $b$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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.

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

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

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

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

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

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

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

Multiply $a$ by $1$ .

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

Step 5

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

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

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

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

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

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

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

Simplify (1+(b^2)/a)/(a/2-a/b)

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