Basic Math Examples

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

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

Step 1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 3

Combine

a^{3}

$a^{3}$ and

\frac{1}{b^{2}}

$\frac{1}{b^{2}}$ .

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

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

Step 4

Combine

a^{4}

$a^{4}$ and

\frac{1}{b^{3}}

$\frac{1}{b^{3}}$ .

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

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 6

Combine.

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

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

Step 7

Cancel the common factor of

$a^{3}$ and

$a^{4}$ .

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Step 7.1

Factor

$a^{3}$ out of

$a^{3} b^{3}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor

$a^{3}$ out of

$b^{2} a^{4}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Cancel the common factor of

$b^{3}$ and

$b^{2}$ .

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Step 8.1

Factor

$b^{2}$ out of

$b^{3}$ .

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor

$b^{2}$ out of

$b^{2} a$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.1

Apply the product rule to

$- \frac{5 a}{2 b}$ .

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

Step 9.2

Apply the product rule to

$\frac{5 a}{2 b}$ .

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

Step 9.3

Apply the product rule to

$5 a$ .

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

Step 9.4

Apply the product rule to

$2 b$ .

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

Step 10

Simplify the expression.

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Step 10.1

Rewrite using the commutative property of multiplication.

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

Step 10.2

Raise

$- 1$ to the power of

$3$ .

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

Step 11

Cancel the common factor of

$b$ .

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Step 11.1

Move the leading negative in

$- \frac{b}{a}$ into the numerator.

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

Step 11.2

Factor

$b$ out of

$- b$ .

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

Step 11.3

Factor

$b$ out of

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

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

Step 11.4

Cancel the common factor.

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

Step 11.5

Rewrite the expression.

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

Step 12

Cancel the common factor of

$a$ .

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Step 12.1

Factor

$a$ out of

$5^{3} a^{3}$ .

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

Evaluate the exponents.

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Step 13.1

Raise

$5$ to the power of

$3$ .

$- \frac{125 a^{2}}{2^{3} b^{2}}$

Step 13.2

Raise

$2$ to the power of

$3$ .

$- \frac{125 a^{2}}{8 b^{2}}$