Simplify a/(3b)-2(a/b-b/(2a))

$\frac{a}{3 b} - 2 (\frac{a}{b} - \frac{b}{2 a})$

Step 1

Simplify each term.

To write $\frac{a}{b}$ as a fraction with a common denominator, multiply by $\frac{2 a}{2 a}$ .

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

To write $- \frac{b}{2 a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Write each expression with a common denominator of $b \cdot 2 a$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $b (2 a)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{a}{3 b} - 2 \frac{2 a \cdot a - b \cdot b}{2 a b}$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

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

Multiply $b$ by $b$ .

$\frac{a}{3 b} - 2 \frac{2 a^{2} - b^{2}}{2 a b}$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Rewrite $- 1 \frac{2 a^{2} - b^{2}}{a b}$ as $- \frac{2 a^{2} - b^{2}}{a b}$ .

$\frac{a}{3 b} - \frac{2 a^{2} - b^{2}}{a b}$

Step 2

To write $\frac{a}{3 b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 3

To write $- \frac{2 a^{2} - b^{2}}{a b}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $3 b a$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a}{3 b}$ by $\frac{a}{a}$ .

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

Multiply $\frac{2 a^{2} - b^{2}}{a b}$ by $\frac{3}{3}$ .

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

Reorder the factors of $3 b a$ .

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

Reorder the factors of $a b \cdot 3$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Multiply $a$ by $a$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- - b^{2}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $b^{2}$ by $1$ .

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

Apply the distributive property.

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

Multiply $3$ by $- 2$ .

$\frac{a^{2} - 6 a^{2} + b^{2} \cdot 3}{3 a b}$

Move $3$ to the left of $b^{2}$ .

$\frac{a^{2} - 6 a^{2} + 3 \cdot b^{2}}{3 a b}$

Subtract $6 a^{2}$ from $a^{2}$ .

$\frac{- 5 a^{2} + 3 b^{2}}{3 a b}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- 5 a^{2}$ .

$\frac{- (5 a^{2}) + 3 b^{2}}{3 a b}$

Factor $- 1$ out of $3 b^{2}$ .

$\frac{- (5 a^{2}) - (- 3 b^{2})}{3 a b}$

Factor $- 1$ out of $- (5 a^{2}) - (- 3 b^{2})$ .

$\frac{- (5 a^{2} - 3 b^{2})}{3 a b}$

Simplify the expression.

Rewrite $- (5 a^{2} - 3 b^{2})$ as $- 1 (5 a^{2} - 3 b^{2})$ .

$\frac{- 1 (5 a^{2} - 3 b^{2})}{3 a b}$

Move the negative in front of the fraction.

$- \frac{5 a^{2} - 3 b^{2}}{3 a b}$

Simplify a/(3b)-2(a/b-b/(2a))

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