Simplify ((1+3a^-1)/(2-a^-1))((a^-2-2a^-1)/(3a^-3+a^-2))

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

Simplify the numerator.

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Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Combine $3$ and $\frac{1}{a}$ .

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the denominator.

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Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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Multiply by $1$ .

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

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

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

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

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Simplify the denominator.

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Factor $a^{- 3}$ out of $3 a^{- 3} + a^{- 2}$ .

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Factor $a^{- 3}$ out of $3 a^{- 3}$ .

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

Factor $a^{- 3}$ out of $a^{- 2}$ .

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

Factor $a^{- 3}$ out of $a^{- 3} \cdot 3 + a^{- 3} a$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $a$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Multiply $a + 3$ and $\frac{1}{2 a - 1}$ .

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

Simplify.

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Factor $\frac{1}{a^{2}}$ out of $\frac{\frac{1}{a^{2}} (1 - 2 a)}{\frac{1}{a^{3}} (3 + a)}$ .

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

Factor $\frac{1}{a^{3}}$ out of $\frac{1 - 2 a}{\frac{1}{a^{3}} (3 + a)}$ .

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

Cancel the common factor of $a^{2}$ .

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Factor $a^{2}$ out of $a^{3} \frac{1 - 2 a}{3 + a}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $a$ and $\frac{1 - 2 a}{3 + a}$ .

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

Multiply $\frac{a + 3}{2 a - 1}$ and $\frac{a (1 - 2 a)}{3 + a}$ .

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

Cancel the common factor of $a + 3$ and $3 + a$ .

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Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{a (1 - 2 a)}{2 a - 1}$

Cancel the common factor of $1 - 2 a$ and $2 a - 1$ .

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Rewrite $1$ as $- 1 (- 1)$ .

$\frac{a (- 1 (- 1) - 2 a)}{2 a - 1}$

Factor $- 1$ out of $- 2 a$ .

$\frac{a (- 1 (- 1) - (2 a))}{2 a - 1}$

Factor $- 1$ out of $- 1 (- 1) - (2 a)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Divide $a \cdot - 1$ by $1$ .

$a \cdot - 1$

Simplify the expression.

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Move $- 1$ to the left of $a$ .

$- 1 \cdot a$

Rewrite $- 1 a$ as $- a$ .

$- a$

Simplify ((1+3a^-1)/(2-a^-1))((a^-2-2a^-1)/(3a^-3+a^-2))

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