Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Multiply by $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

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

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

Step 5

Combine fractions.

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Step 5.1

Combine.

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

Step 5.2

Multiply $\frac{1}{a^{2}}$ by $\frac{a + 3}{a}$ .

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

Step 5.3

Multiply $\frac{1}{a^{3}}$ by $\frac{2 a - 1}{a}$ .

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Add $2$ and $1$ .

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

Step 7

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 7.1

Multiply $a^{3}$ by $a$ .

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Add $3$ and $1$ .

$\frac{\frac{a + 3}{a^{3}} (1 - 2 a)}{\frac{2 a - 1}{a^{4}} (3 + a)}$

Step 8

Factor $\frac{a + 3}{a^{3}}$ out of $\frac{\frac{a + 3}{a^{3}} (1 - 2 a)}{\frac{2 a - 1}{a^{4}} (3 + a)}$ .

$\frac{a + 3}{a^{3}} \cdot \frac{1 - 2 a}{\frac{2 a - 1}{a^{4}} (3 + a)}$

Step 9

Factor $\frac{2 a - 1}{a^{4}}$ out of $\frac{1 - 2 a}{\frac{2 a - 1}{a^{4}} (3 + a)}$ .

$\frac{a + 3}{a^{3}} (\frac{a^{4}}{2 a - 1} \cdot \frac{1 - 2 a}{3 + a})$

Step 10

Multiply $\frac{a^{4}}{2 a - 1}$ by $\frac{1 - 2 a}{3 + a}$ .

$\frac{a + 3}{a^{3}} \cdot \frac{a^{4} (1 - 2 a)}{(2 a - 1) (3 + a)}$

Step 11

Cancel the common factor of $a^{3}$ .

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Step 11.1

Factor $a^{3}$ out of $a^{4} (1 - 2 a)$ .

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

Step 11.2

Cancel the common factor.

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

Step 11.3

Rewrite the expression.

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Reorder terms.

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

Step 12.5

Cancel the common factor.

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

Step 12.6

Rewrite the expression.

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

Step 13

Simplify the expression.

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Step 13.1

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

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

Step 13.2

Move the negative in front of the fraction.

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

Step 14

Apply the distributive property.

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

Step 15

Rewrite using the commutative property of multiplication.

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

Step 16

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

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Step 16.1

Multiply $- 1$ by $3$ .

$- a \frac{a}{3 + a} - 3 \frac{a}{3 + a}$

Step 16.2

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

$- a \frac{a}{3 + a} + \frac{- 3 a}{3 + a}$

Step 17

Simplify each term.

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Step 17.1

Multiply $- a \frac{a}{3 + a}$ .

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Step 17.1.1

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

$- \frac{a \cdot a}{3 + a} + \frac{- 3 a}{3 + a}$

Step 17.1.2

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

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

Step 17.1.3

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

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

Step 17.1.4

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

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

Step 17.1.5

Add $1$ and $1$ .

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

Step 17.2

Move the negative in front of the fraction.

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

Step 18

Simplify terms.

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Step 18.1

Combine the numerators over the common denominator.

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

Step 18.2

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

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Step 18.2.1

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

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

Step 18.2.2

Factor $a$ out of $- 3 a$ .

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

Step 18.2.3

Factor $a$ out of $a (- a) + a \cdot - 3$ .

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

Step 18.3

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

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Step 18.3.1

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

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

Step 18.3.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 18.3.3

Factor $- 1$ out of $- (a) - 1 (3)$ .

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

Step 18.3.4

Rewrite $- (a + 3)$ as $- 1 (a + 3)$ .

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

Step 18.3.5

Reorder terms.

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

Step 18.3.6

Cancel the common factor.

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

Step 18.3.7

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

$a \cdot (- 1)$

Step 18.4

Simplify the expression.

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Step 18.4.1

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

$- 1 a$

Step 18.4.2

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

$- a$