Pre-Algebra Examples

$\frac{20 a^{6} b^{3}}{4 a^{6} b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $20$ and $4$ .

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Step 1.1.1

Factor $4$ out of $20 a^{6} b^{3}$ .

$\frac{4 (5 a^{6} b^{3})}{4 a^{6} b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $4$ out of $4 a^{6} b$ .

$\frac{4 (5 a^{6} b^{3})}{4 (a^{6} b)} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.1.2.2

Cancel the common factor.

$\frac{4 (5 a^{6} b^{3})}{4 (a^{6} b)} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.1.2.3

Rewrite the expression.

$\frac{5 a^{6} b^{3}}{a^{6} b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.2

Cancel the common factor of $a^{6}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{5 a^{6} b^{3}}{a^{6} b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.2.2

Rewrite the expression.

$\frac{5 b^{3}}{b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3

Cancel the common factor of $b^{3}$ and $b$ .

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Step 1.3.1

Factor $b$ out of $5 b^{3}$ .

$\frac{b (5 b^{2})}{b} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{b (5 b^{2})}{b^{1}} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3.2.2

Factor $b$ out of $b^{1}$ .

$\frac{b (5 b^{2})}{b \cdot 1} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3.2.3

Cancel the common factor.

$\frac{b (5 b^{2})}{b \cdot 1} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3.2.4

Rewrite the expression.

$\frac{5 b^{2}}{1} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.3.2.5

Divide $5 b^{2}$ by $1$ .

$5 b^{2} - \frac{16 a^{9} b^{2}}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.4

Cancel the common factor of $16$ and $4$ .

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Step 1.4.1

Factor $4$ out of $16 a^{9} b^{2}$ .

$5 b^{2} - \frac{4 (4 a^{9} b^{2})}{4 a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $4$ out of $4 a^{6} b$ .

$5 b^{2} - \frac{4 (4 a^{9} b^{2})}{4 (a^{6} b)} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.4.2.2

Cancel the common factor.

$5 b^{2} - \frac{4 (4 a^{9} b^{2})}{4 (a^{6} b)} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.4.2.3

Rewrite the expression.

$5 b^{2} - \frac{4 a^{9} b^{2}}{a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.5

Cancel the common factor of $a^{9}$ and $a^{6}$ .

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Step 1.5.1

Factor $a^{6}$ out of $4 a^{9} b^{2}$ .

$5 b^{2} - \frac{a^{6} (4 a^{3} b^{2})}{a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $a^{6}$ out of $a^{6} b$ .

$5 b^{2} - \frac{a^{6} (4 a^{3} b^{2})}{a^{6} (b)} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.5.2.2

Cancel the common factor.

$5 b^{2} - \frac{a^{6} (4 a^{3} b^{2})}{a^{6} b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.5.2.3

Rewrite the expression.

$5 b^{2} - \frac{4 a^{3} b^{2}}{b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6

Cancel the common factor of $b^{2}$ and $b$ .

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Step 1.6.1

Factor $b$ out of $4 a^{3} b^{2}$ .

$5 b^{2} - \frac{b (4 a^{3} b)}{b} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6.2

Cancel the common factors.

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Step 1.6.2.1

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

$5 b^{2} - \frac{b (4 a^{3} b)}{b^{1}} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6.2.2

Factor $b$ out of $b^{1}$ .

$5 b^{2} - \frac{b (4 a^{3} b)}{b \cdot 1} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6.2.3

Cancel the common factor.

$5 b^{2} - \frac{b (4 a^{3} b)}{b \cdot 1} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6.2.4

Rewrite the expression.

$5 b^{2} - \frac{4 a^{3} b}{1} + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.6.2.5

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

$5 b^{2} - (4 a^{3} b) + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.7

Multiply $4$ by $- 1$ .

$5 b^{2} - 4 (a^{3} b) + \frac{8 a^{2} b}{4 a^{6} b}$

Step 1.8

Cancel the common factor of $8$ and $4$ .

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Step 1.8.1

Factor $4$ out of $8 a^{2} b$ .

$5 b^{2} - 4 a^{3} b + \frac{4 (2 a^{2} b)}{4 a^{6} b}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $4$ out of $4 a^{6} b$ .

$5 b^{2} - 4 a^{3} b + \frac{4 (2 a^{2} b)}{4 (a^{6} b)}$

Step 1.8.2.2

Cancel the common factor.

$5 b^{2} - 4 a^{3} b + \frac{4 (2 a^{2} b)}{4 (a^{6} b)}$

Step 1.8.2.3

Rewrite the expression.

$5 b^{2} - 4 a^{3} b + \frac{2 a^{2} b}{a^{6} b}$

Step 1.9

Cancel the common factor of $a^{2}$ and $a^{6}$ .

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Step 1.9.1

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

$5 b^{2} - 4 a^{3} b + \frac{a^{2} (2 b)}{a^{6} b}$

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $a^{2}$ out of $a^{6} b$ .

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

Step 1.9.2.2

Cancel the common factor.

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

Step 1.9.2.3

Rewrite the expression.

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

Step 1.10

Cancel the common factor of $b$ .

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Step 1.10.1

Cancel the common factor.

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

Step 1.10.2

Rewrite the expression.

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

Step 2

To write $5 b^{2}$ as a fraction with a common denominator, multiply by $\frac{a^{4}}{a^{4}}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

To write $- 4 a^{3} b$ as a fraction with a common denominator, multiply by $\frac{a^{4}}{a^{4}}$ .

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

Step 5

Combine $- 4 a^{3} b$ and $\frac{a^{4}}{a^{4}}$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Multiply $a^{3}$ by $a^{4}$ by adding the exponents.

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Step 7.1

Move $a^{4}$ .

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

Step 7.2

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

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

Step 7.3

Add $4$ and $3$ .

$\frac{- 4 a^{7} b + 5 b^{2} a^{4} + 2}{a^{4}}$

Step 8

Factor $- 1$ out of $- 4 a^{7} b$ .

$\frac{- (4 a^{7} b) + 5 b^{2} a^{4} + 2}{a^{4}}$

Step 9

Factor $- 1$ out of $5 b^{2} a^{4}$ .

$\frac{- (4 a^{7} b) - (- 5 b^{2} a^{4}) + 2}{a^{4}}$

Step 10

Factor $- 1$ out of $- (4 a^{7} b) - (- 5 b^{2} a^{4})$ .

$\frac{- (4 a^{7} b - 5 b^{2} a^{4}) + 2}{a^{4}}$

Step 11

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (4 a^{7} b - 5 b^{2} a^{4}) - 1 (- 2)}{a^{4}}$

Step 12

Factor $- 1$ out of $- (4 a^{7} b - 5 b^{2} a^{4}) - 1 (- 2)$ .

$\frac{- (4 a^{7} b - 5 b^{2} a^{4} - 2)}{a^{4}}$

Step 13

Simplify the expression.

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Step 13.1

Rewrite $- (4 a^{7} b - 5 b^{2} a^{4} - 2)$ as $- 1 (4 a^{7} b - 5 b^{2} a^{4} - 2)$ .

$\frac{- 1 (4 a^{7} b - 5 b^{2} a^{4} - 2)}{a^{4}}$

Step 13.2

Move the negative in front of the fraction.

$- \frac{4 a^{7} b - 5 b^{2} a^{4} - 2}{a^{4}}$