Pre-Algebra Examples

$\frac{2 a^{4} - \frac{1}{5} a^{2} - \frac{2}{3} a}{- \frac{1}{2} a}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a$ out of $2 a^{4} - \frac{1}{5} a^{2} - \frac{2}{3} a$ .

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Step 1.1.1

Factor $a$ out of $2 a^{4}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.2

Combine $a$ and $\frac{1}{5}$ .

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

Step 1.3

To write $2 a^{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.4

Combine $2 a^{3}$ and $\frac{5}{5}$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Factor $a$ out of $2 a^{3} \cdot 5 - a$ .

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Step 1.6.1.1

Factor $a$ out of $2 a^{3} \cdot 5$ .

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

Step 1.6.1.2

Factor $a$ out of $- a$ .

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

Step 1.6.1.3

Factor $a$ out of $a (2 a^{2} \cdot 5) + a \cdot - 1$ .

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

Step 1.6.2

Multiply $5$ by $2$ .

$\frac{a (\frac{a (10 a^{2} - 1)}{5} - \frac{2}{3})}{- \frac{1}{2} a}$

Step 1.7

To write $\frac{a (10 a^{2} - 1)}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{a (\frac{a (10 a^{2} - 1)}{5} \cdot \frac{3}{3} - \frac{2}{3})}{- \frac{1}{2} a}$

Step 1.8

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{a (\frac{a (10 a^{2} - 1)}{5} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{5}{5})}{- \frac{1}{2} a}$

Step 1.9

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.9.1

Multiply $\frac{a (10 a^{2} - 1)}{5}$ by $\frac{3}{3}$ .

$\frac{a (\frac{a (10 a^{2} - 1) \cdot 3}{5 \cdot 3} - \frac{2}{3} \cdot \frac{5}{5})}{- \frac{1}{2} a}$

Step 1.9.2

Multiply $5$ by $3$ .

$\frac{a (\frac{a (10 a^{2} - 1) \cdot 3}{15} - \frac{2}{3} \cdot \frac{5}{5})}{- \frac{1}{2} a}$

Step 1.9.3

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$\frac{a (\frac{a (10 a^{2} - 1) \cdot 3}{15} - \frac{2 \cdot 5}{3 \cdot 5})}{- \frac{1}{2} a}$

Step 1.9.4

Multiply $3$ by $5$ .

$\frac{a (\frac{a (10 a^{2} - 1) \cdot 3}{15} - \frac{2 \cdot 5}{15})}{- \frac{1}{2} a}$

Step 1.10

Combine the numerators over the common denominator.

$\frac{a \frac{a (10 a^{2} - 1) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11

Simplify the numerator.

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Step 1.11.1

Apply the distributive property.

$\frac{a \frac{(a (10 a^{2}) + a \cdot - 1) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.2

Rewrite using the commutative property of multiplication.

$\frac{a \frac{(10 a \cdot a^{2} + a \cdot - 1) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.3

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

$\frac{a \frac{(10 a \cdot a^{2} - 1 \cdot a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.4

Simplify each term.

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Step 1.11.4.1

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

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Step 1.11.4.1.1

Move $a^{2}$ .

$\frac{a \frac{(10 (a^{2} a) - 1 \cdot a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.4.1.2

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

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Step 1.11.4.1.2.1

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

$\frac{a \frac{(10 (a^{2} a^{1}) - 1 \cdot a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.4.1.2.2

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

$\frac{a \frac{(10 a^{2 + 1} - 1 \cdot a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.4.1.3

Add $2$ and $1$ .

$\frac{a \frac{(10 a^{3} - 1 \cdot a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.4.2

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

$\frac{a \frac{(10 a^{3} - a) \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.5

Apply the distributive property.

$\frac{a \frac{10 a^{3} \cdot 3 - a \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.6

Multiply $3$ by $10$ .

$\frac{a \frac{30 a^{3} - a \cdot 3 - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.7

Multiply $3$ by $- 1$ .

$\frac{a \frac{30 a^{3} - 3 a - 2 \cdot 5}{15}}{- \frac{1}{2} a}$

Step 1.11.8

Multiply $- 2$ by $5$ .

$\frac{a \frac{30 a^{3} - 3 a - 10}{15}}{- \frac{1}{2} a}$

Step 2

Combine $a$ and $\frac{1}{2}$ .

$\frac{a \frac{30 a^{3} - 3 a - 10}{15}}{- \frac{a}{2}}$

Step 3

Combine $a$ and $\frac{30 a^{3} - 3 a - 10}{15}$ .

$\frac{\frac{a (30 a^{3} - 3 a - 10)}{15}}{- \frac{a}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{a (30 a^{3} - 3 a - 10)}{15} (- \frac{2}{a})$

Step 5

Rewrite using the commutative property of multiplication.

$- \frac{a (30 a^{3} - 3 a - 10)}{15} \cdot \frac{2}{a}$

Step 6

Cancel the common factor of $a$ .

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Step 6.1

Move the leading negative in $- \frac{a (30 a^{3} - 3 a - 10)}{15}$ into the numerator.

$\frac{- a (30 a^{3} - 3 a - 10)}{15} \cdot \frac{2}{a}$

Step 6.2

Factor $a$ out of $- a (30 a^{3} - 3 a - 10)$ .

$\frac{a (- (30 a^{3} - 3 a - 10))}{15} \cdot \frac{2}{a}$

Step 6.3

Cancel the common factor.

$\frac{a (- (30 a^{3} - 3 a - 10))}{15} \cdot \frac{2}{a}$

Step 6.4

Rewrite the expression.

$\frac{- (30 a^{3} - 3 a - 10)}{15} \cdot 2$

Step 7

Combine $\frac{- (30 a^{3} - 3 a - 10)}{15}$ and $2$ .

$\frac{- (30 a^{3} - 3 a - 10) \cdot 2}{15}$

Step 8

Multiply $2$ by $- 1$ .

$\frac{- 2 (30 a^{3} - 3 a - 10)}{15}$

Step 9

Move the negative in front of the fraction.

$- \frac{2 (30 a^{3} - 3 a - 10)}{15}$

Step 10

Apply the distributive property.

$- \frac{2 (30 a^{3}) + 2 (- 3 a) + 2 \cdot - 10}{15}$

Step 11

Simplify.

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Step 11.1

Multiply $30$ by $2$ .

$- \frac{60 a^{3} + 2 (- 3 a) + 2 \cdot - 10}{15}$

Step 11.2

Multiply $- 3$ by $2$ .

$- \frac{60 a^{3} - 6 a + 2 \cdot - 10}{15}$

Step 11.3

Multiply $2$ by $- 10$ .

$- \frac{60 a^{3} - 6 a - 20}{15}$

Step 12

Split the fraction $\frac{60 a^{3} - 6 a - 20}{15}$ into two fractions.

$- (\frac{60 a^{3} - 6 a}{15} + \frac{- 20}{15})$

Step 13

Split the fraction $\frac{60 a^{3} - 6 a}{15}$ into two fractions.

$- (\frac{60 a^{3}}{15} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 14

Cancel the common factor of $60$ and $15$ .

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Step 14.1

Factor $15$ out of $60 a^{3}$ .

$- (\frac{15 (4 a^{3})}{15} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $15$ out of $15$ .

$- (\frac{15 (4 a^{3})}{15 (1)} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 14.2.2

Cancel the common factor.

$- (\frac{15 (4 a^{3})}{15 \cdot 1} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 14.2.3

Rewrite the expression.

$- (\frac{4 a^{3}}{1} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 14.2.4

Divide $4 a^{3}$ by $1$ .

$- (4 a^{3} + \frac{- 6 a}{15} + \frac{- 20}{15})$

Step 15

Cancel the common factor of $- 6$ and $15$ .

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Step 15.1

Factor $3$ out of $- 6 a$ .

$- (4 a^{3} + \frac{3 (- 2 a)}{15} + \frac{- 20}{15})$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $3$ out of $15$ .

$- (4 a^{3} + \frac{3 (- 2 a)}{3 (5)} + \frac{- 20}{15})$

Step 15.2.2

Cancel the common factor.

$- (4 a^{3} + \frac{3 (- 2 a)}{3 \cdot 5} + \frac{- 20}{15})$

Step 15.2.3

Rewrite the expression.

$- (4 a^{3} + \frac{- 2 a}{5} + \frac{- 20}{15})$

Step 16

Move the negative in front of the fraction.

$- (4 a^{3} - \frac{2 a}{5} + \frac{- 20}{15})$

Step 17

Cancel the common factor of $- 20$ and $15$ .

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Step 17.1

Factor $5$ out of $- 20$ .

$- (4 a^{3} - \frac{2 a}{5} + \frac{5 (- 4)}{15})$

Step 17.2

Cancel the common factors.

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Step 17.2.1

Factor $5$ out of $15$ .

$- (4 a^{3} - \frac{2 a}{5} + \frac{5 \cdot - 4}{5 \cdot 3})$

Step 17.2.2

Cancel the common factor.

$- (4 a^{3} - \frac{2 a}{5} + \frac{5 \cdot - 4}{5 \cdot 3})$

Step 17.2.3

Rewrite the expression.

$- (4 a^{3} - \frac{2 a}{5} + \frac{- 4}{3})$

Step 18

Move the negative in front of the fraction.

$- (4 a^{3} - \frac{2 a}{5} - \frac{4}{3})$