Basic Math Examples

$- \frac{6}{a^{2} b^{2}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 1

To write $- \frac{6}{a^{2} b^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2} b^{4}}{a^{2} b^{4}}$ .

$- \frac{6}{a^{2} b^{2}} \cdot \frac{a^{2} b^{4}}{a^{2} b^{4}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2

Write each expression with a common denominator of $a^{4} b^{6}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{6}{a^{2} b^{2}}$ by $\frac{a^{2} b^{4}}{a^{2} b^{4}}$ .

$- \frac{6 (a^{2} b^{4})}{a^{2} b^{2} (a^{2} b^{4})} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2.2

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

$- \frac{6 (a^{2} b^{4})}{a^{2 + 2} b^{2} b^{4}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2.3

Add $2$ and $2$ .

$- \frac{6 (a^{2} b^{4})}{a^{4} b^{2} b^{4}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2.4

Multiply $b^{2}$ by $b^{4}$ by adding the exponents.

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Step 2.4.1

Move $b^{4}$ .

$- \frac{6 (a^{2} b^{4})}{a^{4} (b^{4} b^{2})} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2.4.2

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

$- \frac{6 (a^{2} b^{4})}{a^{4} b^{4 + 2}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 2.4.3

Add $4$ and $2$ .

$- \frac{6 (a^{2} b^{4})}{a^{4} b^{6}} + \frac{2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 6 (a^{2} b^{4}) + 2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 4

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

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Step 4.1

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

$\frac{2 (- 3 a^{2} b^{4}) + 2}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 4.2

Factor $2$ out of $2$ .

$\frac{2 (- 3 a^{2} b^{4}) + 2 (1)}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 4.3

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

$\frac{2 (- 3 a^{2} b^{4} + 1)}{a^{4} b^{6}} - \frac{8}{a^{6} b^{7}}$

Step 5

To write $\frac{2 (- 3 a^{2} b^{4} + 1)}{a^{4} b^{6}}$ as a fraction with a common denominator, multiply by $\frac{a^{2} b}{a^{2} b}$ .

$\frac{2 (- 3 a^{2} b^{4} + 1)}{a^{4} b^{6}} \cdot \frac{a^{2} b}{a^{2} b} - \frac{8}{a^{6} b^{7}}$

Step 6

Write each expression with a common denominator of $a^{6} b^{7}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{2 (- 3 a^{2} b^{4} + 1)}{a^{4} b^{6}}$ by $\frac{a^{2} b}{a^{2} b}$ .

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{4} b^{6} (a^{2} b)} - \frac{8}{a^{6} b^{7}}$

Step 6.2

Multiply $a^{4}$ by $a^{2}$ by adding the exponents.

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Step 6.2.1

Move $a^{2}$ .

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{2} a^{4} b^{6} b} - \frac{8}{a^{6} b^{7}}$

Step 6.2.2

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

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{2 + 4} b^{6} b} - \frac{8}{a^{6} b^{7}}$

Step 6.2.3

Add $2$ and $4$ .

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{6} b^{6} b} - \frac{8}{a^{6} b^{7}}$

Step 6.3

Multiply $b^{6}$ by $b$ by adding the exponents.

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Step 6.3.1

Move $b$ .

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{6} (b \cdot b^{6})} - \frac{8}{a^{6} b^{7}}$

Step 6.3.2

Multiply $b$ by $b^{6}$ .

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Step 6.3.2.1

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

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{6} (b^{1} b^{6})} - \frac{8}{a^{6} b^{7}}$

Step 6.3.2.2

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

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{6} b^{1 + 6}} - \frac{8}{a^{6} b^{7}}$

Step 6.3.3

Add $1$ and $6$ .

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b)}{a^{6} b^{7}} - \frac{8}{a^{6} b^{7}}$

Step 7

Combine the numerators over the common denominator.

$\frac{2 (- 3 a^{2} b^{4} + 1) (a^{2} b) - 8}{a^{6} b^{7}}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $2$ out of $2 (- 3 a^{2} b^{4} + 1) (a^{2} b) - 8$ .

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Step 8.1.1

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

$\frac{2 ((- 3 a^{2} b^{4} + 1) (a^{2} b)) - 8}{a^{6} b^{7}}$

Step 8.1.2

Factor $2$ out of $- 8$ .

$\frac{2 ((- 3 a^{2} b^{4} + 1) (a^{2} b)) + 2 \cdot - 4}{a^{6} b^{7}}$

Step 8.1.3

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

$\frac{2 ((- 3 a^{2} b^{4} + 1) (a^{2} b) - 4)}{a^{6} b^{7}}$

Step 8.2

Apply the distributive property.

$\frac{2 (- 3 a^{2} b^{4} (a^{2} b) + 1 (a^{2} b) - 4)}{a^{6} b^{7}}$

Step 8.3

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

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Step 8.3.1

Move $a^{2}$ .

$\frac{2 (- 3 (a^{2} a^{2}) b^{4} b + 1 (a^{2} b) - 4)}{a^{6} b^{7}}$

Step 8.3.2

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

$\frac{2 (- 3 a^{2 + 2} b^{4} b + 1 (a^{2} b) - 4)}{a^{6} b^{7}}$

Step 8.3.3

Add $2$ and $2$ .

$\frac{2 (- 3 a^{4} b^{4} b + 1 (a^{2} b) - 4)}{a^{6} b^{7}}$

Step 8.4

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

$\frac{2 (- 3 a^{4} b^{4} b + a^{2} b - 4)}{a^{6} b^{7}}$

Step 8.5

Multiply $b^{4}$ by $b$ by adding the exponents.

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Step 8.5.1

Move $b$ .

$\frac{2 (- 3 a^{4} (b \cdot b^{4}) + a^{2} b - 4)}{a^{6} b^{7}}$

Step 8.5.2

Multiply $b$ by $b^{4}$ .

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Step 8.5.2.1

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

$\frac{2 (- 3 a^{4} (b^{1} b^{4}) + a^{2} b - 4)}{a^{6} b^{7}}$

Step 8.5.2.2

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

$\frac{2 (- 3 a^{4} b^{1 + 4} + a^{2} b - 4)}{a^{6} b^{7}}$

Step 8.5.3

Add $1$ and $4$ .

$\frac{2 (- 3 a^{4} b^{5} + a^{2} b - 4)}{a^{6} b^{7}}$

Step 9

Simplify with factoring out.

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Step 9.1

Factor $- 1$ out of $- 3 a^{4} b^{5}$ .

$\frac{2 (- (3 a^{4} b^{5}) + a^{2} b - 4)}{a^{6} b^{7}}$

Step 9.2

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

$\frac{2 (- (3 a^{4} b^{5}) - (- a^{2} b) - 4)}{a^{6} b^{7}}$

Step 9.3

Factor $- 1$ out of $- (3 a^{4} b^{5}) - (- a^{2} b)$ .

$\frac{2 (- (3 a^{4} b^{5} - a^{2} b) - 4)}{a^{6} b^{7}}$

Step 9.4

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{2 (- (3 a^{4} b^{5} - a^{2} b) - 1 (4))}{a^{6} b^{7}}$

Step 9.5

Factor $- 1$ out of $- (3 a^{4} b^{5} - a^{2} b) - 1 (4)$ .

$\frac{2 (- (3 a^{4} b^{5} - a^{2} b + 4))}{a^{6} b^{7}}$

Step 9.6

Simplify the expression.

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Step 9.6.1

Rewrite $- (3 a^{4} b^{5} - a^{2} b + 4)$ as $- 1 (3 a^{4} b^{5} - a^{2} b + 4)$ .

$\frac{2 (- 1 (3 a^{4} b^{5} - a^{2} b + 4))}{a^{6} b^{7}}$

Step 9.6.2

Move the negative in front of the fraction.

$- \frac{2 (3 a^{4} b^{5} - a^{2} b + 4)}{a^{6} b^{7}}$