Finite Math Examples

$\frac{5^{\frac{3}{2}} b^{\frac{3}{2}} \cdot (81 a^{6})}{{(5 a)}^{\frac{3}{2}} {(9 b)}^{4}}$

Step 1

Simplify the denominator.

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Step 1.1

Apply the product rule to $9 b$ .

$\frac{5^{\frac{3}{2}} b^{\frac{3}{2}} \cdot 81 a^{6}}{{(5 a)}^{\frac{3}{2}} \cdot 9^{4} b^{4}}$

Step 1.2

Apply the product rule to $5 a$ .

$\frac{5^{\frac{3}{2}} b^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 9^{4} b^{4}}$

Step 1.3

Raise $9$ to the power of $4$ .

$\frac{5^{\frac{3}{2}} b^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{4}}$

Step 2

Move $b^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{4} b^{- \frac{3}{2}}}$

Step 3

Multiply $b^{4}$ by $b^{- \frac{3}{2}}$ by adding the exponents.

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Step 3.1

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

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 (b^{- \frac{3}{2}} b^{4})}$

Step 3.2

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

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{- \frac{3}{2} + 4}}$

Step 3.3

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

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{- \frac{3}{2} + 4 \cdot \frac{2}{2}}}$

Step 3.4

Combine $4$ and $\frac{2}{2}$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{- \frac{3}{2} + \frac{4 \cdot 2}{2}}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{\frac{- 3 + 4 \cdot 2}{2}}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $4$ by $2$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{\frac{- 3 + 8}{2}}}$

Step 3.6.2

Add $- 3$ and $8$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6}}{5^{\frac{3}{2}} a^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 4

Move $a^{\frac{3}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{6} a^{- \frac{3}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5

Multiply $a^{6}$ by $a^{- \frac{3}{2}}$ by adding the exponents.

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Step 5.1

Move $a^{- \frac{3}{2}}$ .

$\frac{5^{\frac{3}{2}} \cdot 81 (a^{- \frac{3}{2}} a^{6})}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.2

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

$\frac{5^{\frac{3}{2}} \cdot 81 a^{- \frac{3}{2} + 6}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.3

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

$\frac{5^{\frac{3}{2}} \cdot 81 a^{- \frac{3}{2} + 6 \cdot \frac{2}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.4

Combine $6$ and $\frac{2}{2}$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{- \frac{3}{2} + \frac{6 \cdot 2}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{5^{\frac{3}{2}} \cdot 81 a^{\frac{- 3 + 6 \cdot 2}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $6$ by $2$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{\frac{- 3 + 12}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 5.6.2

Add $- 3$ and $12$ .

$\frac{5^{\frac{3}{2}} \cdot 81 a^{\frac{9}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 6

Cancel the common factor.

$\frac{5^{\frac{3}{2}} \cdot 81 a^{\frac{9}{2}}}{5^{\frac{3}{2}} \cdot 6561 b^{\frac{5}{2}}}$

Step 7

Rewrite the expression.

$\frac{81 a^{\frac{9}{2}}}{6561 b^{\frac{5}{2}}}$

Step 8

Factor $81$ out of $81 a^{\frac{9}{2}}$ .

$\frac{81 (a^{\frac{9}{2}})}{6561 b^{\frac{5}{2}}}$

Step 9

Cancel the common factors.

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Step 9.1

Factor $81$ out of $6561 b^{\frac{5}{2}}$ .

$\frac{81 (a^{\frac{9}{2}})}{81 (81 b^{\frac{5}{2}})}$

Step 9.2

Cancel the common factor.

$\frac{81 a^{\frac{9}{2}}}{81 (81 b^{\frac{5}{2}})}$

Step 9.3

Rewrite the expression.

$\frac{a^{\frac{9}{2}}}{81 b^{\frac{5}{2}}}$