Basic Math Examples

\frac{- 3}{2 a^{3} b^{2}} - \frac{7}{3 a b^{4}}

$\frac{- 3}{2 a^{3} b^{2}} - \frac{7}{3 a b^{4}}$

Step 1

Move the negative in front of the fraction.

- \frac{3}{2 a^{3} b^{2}} - \frac{7}{3 a b^{4}}

$- \frac{3}{2 a^{3} b^{2}} - \frac{7}{3 a b^{4}}$

Step 2

To write

- \frac{3}{2 a^{3} b^{2}}

$- \frac{3}{2 a^{3} b^{2}}$ as a fraction with a common denominator, multiply by

\frac{3 b^{2}}{3 b^{2}}

$\frac{3 b^{2}}{3 b^{2}}$ .

- \frac{3}{2 a^{3} b^{2}} \cdot \frac{3 b^{2}}{3 b^{2}} - \frac{7}{3 a b^{4}}

$- \frac{3}{2 a^{3} b^{2}} \cdot \frac{3 b^{2}}{3 b^{2}} - \frac{7}{3 a b^{4}}$

Step 3

To write

- \frac{7}{3 a b^{4}}

$- \frac{7}{3 a b^{4}}$ as a fraction with a common denominator, multiply by

\frac{2 a^{2}}{2 a^{2}}

$\frac{2 a^{2}}{2 a^{2}}$ .

- \frac{3}{2 a^{3} b^{2}} \cdot \frac{3 b^{2}}{3 b^{2}} - \frac{7}{3 a b^{4}} \cdot \frac{2 a^{2}}{2 a^{2}}

$- \frac{3}{2 a^{3} b^{2}} \cdot \frac{3 b^{2}}{3 b^{2}} - \frac{7}{3 a b^{4}} \cdot \frac{2 a^{2}}{2 a^{2}}$

Step 4

Write each expression with a common denominator of

6 a^{3} b^{4}

$6 a^{3} b^{4}$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 4.1

Multiply

\frac{3}{2 a^{3} b^{2}}

$\frac{3}{2 a^{3} b^{2}}$ by

\frac{3 b^{2}}{3 b^{2}}

$\frac{3 b^{2}}{3 b^{2}}$ .

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

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

Step 4.2

Multiply

3

$3$ by

2

$2$ .

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

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

Step 4.3

Multiply

b^{2}

$b^{2}$ by

b^{2}

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

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Step 4.3.1

Move

b^{2}

$b^{2}$ .

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

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

Step 4.3.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.3.3

Add

2

$2$ and

2

$2$ .

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

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

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

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

Step 4.4

Multiply

\frac{7}{3 a b^{4}}

$\frac{7}{3 a b^{4}}$ by

\frac{2 a^{2}}{2 a^{2}}

$\frac{2 a^{2}}{2 a^{2}}$ .

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

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

Step 4.5

Multiply

2

$2$ by

3

$3$ .

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

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

Step 4.6

Multiply

a

$a$ by

a^{2}

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

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Step 4.6.1

Move

a^{2}

$a^{2}$ .

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

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

Step 4.6.2

Multiply

a^{2}

$a^{2}$ by

a

$a$ .

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Step 4.6.2.1

Raise

a

$a$ to the power of

1

$1$ .

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

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

Step 4.6.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.6.3

Add

2

$2$ and

1

$1$ .

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

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply

- 3

$- 3$ by

3

$3$ .

\frac{- 9 b^{2} - 7 \cdot 2 a^{2}}{6 a^{3} b^{4}}

$\frac{- 9 b^{2} - 7 \cdot 2 a^{2}}{6 a^{3} b^{4}}$

Step 6.2

Multiply

- 7

$- 7$ by

2

$2$ .

\frac{- 9 b^{2} - 14 a^{2}}{6 a^{3} b^{4}}

$\frac{- 9 b^{2} - 14 a^{2}}{6 a^{3} b^{4}}$

\frac{- 9 b^{2} - 14 a^{2}}{6 a^{3} b^{4}}

$\frac{- 9 b^{2} - 14 a^{2}}{6 a^{3} b^{4}}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor

- 1

$- 1$ out of

- 9 b^{2}

$- 9 b^{2}$ .

\frac{- (9 b^{2}) - 14 a^{2}}{6 a^{3} b^{4}}

$\frac{- (9 b^{2}) - 14 a^{2}}{6 a^{3} b^{4}}$

Step 7.2

Factor

- 1

$- 1$ out of

- 14 a^{2}

$- 14 a^{2}$ .

\frac{- (9 b^{2}) - (14 a^{2})}{6 a^{3} b^{4}}

$\frac{- (9 b^{2}) - (14 a^{2})}{6 a^{3} b^{4}}$

Step 7.3

Factor

- 1

$- 1$ out of

- (9 b^{2}) - (14 a^{2})

$- (9 b^{2}) - (14 a^{2})$ .

\frac{- (9 b^{2} + 14 a^{2})}{6 a^{3} b^{4}}

$\frac{- (9 b^{2} + 14 a^{2})}{6 a^{3} b^{4}}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite

- (9 b^{2} + 14 a^{2})

$- (9 b^{2} + 14 a^{2})$ as

- 1 (9 b^{2} + 14 a^{2})

$- 1 (9 b^{2} + 14 a^{2})$ .

\frac{- 1 (9 b^{2} + 14 a^{2})}{6 a^{3} b^{4}}

$\frac{- 1 (9 b^{2} + 14 a^{2})}{6 a^{3} b^{4}}$

Step 7.4.2

Move the negative in front of the fraction.

- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}

$- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}$

- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}

$- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}$

- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}

$- \frac{9 b^{2} + 14 a^{2}}{6 a^{3} b^{4}}$