Finite Math Examples

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2}{3}} \cdot 9^{- \frac{1}{5}}}$

Step 1

Multiply $9^{\frac{2}{3}}$ by $9^{- \frac{1}{5}}$ by adding the exponents.

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Step 1.1

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

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2}{3} - \frac{1}{5}}}$

Step 1.2

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2}{3} \cdot \frac{5}{5} - \frac{1}{5}}}$

Step 1.3

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2}{3} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{3}{3}}}$

Step 1.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2}{3}$ by $\frac{5}{5}$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2 \cdot 5}{3 \cdot 5} - \frac{1}{5} \cdot \frac{3}{3}}}$

Step 1.4.2

Multiply $3$ by $5$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2 \cdot 5}{15} - \frac{1}{5} \cdot \frac{3}{3}}}$

Step 1.4.3

Multiply $\frac{1}{5}$ by $\frac{3}{3}$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2 \cdot 5}{15} - \frac{3}{5 \cdot 3}}}$

Step 1.4.4

Multiply $5$ by $3$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2 \cdot 5}{15} - \frac{3}{15}}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{9^{- \frac{5}{3}}}{9^{\frac{2 \cdot 5 - 3}{15}}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $2$ by $5$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{10 - 3}{15}}}$

Step 1.6.2

Subtract $3$ from $10$ .

$\frac{9^{- \frac{5}{3}}}{9^{\frac{7}{15}}}$

Step 2

Move $9^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{9^{\frac{7}{15}} \cdot 9^{\frac{5}{3}}}$

Step 3

Multiply $9^{\frac{7}{15}}$ by $9^{\frac{5}{3}}$ by adding the exponents.

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Step 3.1

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

$\frac{1}{9^{\frac{7}{15} + \frac{5}{3}}}$

Step 3.2

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{9^{\frac{7}{15} + \frac{5}{3} \cdot \frac{5}{5}}}$

Step 3.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{5}{3}$ by $\frac{5}{5}$ .

$\frac{1}{9^{\frac{7}{15} + \frac{5 \cdot 5}{3 \cdot 5}}}$

Step 3.3.2

Multiply $3$ by $5$ .

$\frac{1}{9^{\frac{7}{15} + \frac{5 \cdot 5}{15}}}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{9^{\frac{7 + 5 \cdot 5}{15}}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $5$ by $5$ .

$\frac{1}{9^{\frac{7 + 25}{15}}}$

Step 3.5.2

Add $7$ and $25$ .

$\frac{1}{9^{\frac{32}{15}}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{9^{\frac{32}{15}}}$

Decimal Form:

$0.00921048 \dots$