Finite Math Examples

$\frac{81^{- 3}}{9^{- 11}}$

Step 1

Move $81^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{9^{- 11} \cdot 81^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{{(3^{2})}^{- 11} \cdot 81^{3}}$

Step 2.2

Multiply the exponents in ${(3^{2})}^{- 11}$ .

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Step 2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3^{2 \cdot - 11} \cdot 81^{3}}$

Step 2.2.2

Multiply $2$ by $- 11$ .

$\frac{1}{3^{- 22} \cdot 81^{3}}$

Step 2.3

Rewrite $81$ as $3^{4}$ .

$\frac{1}{3^{- 22} \cdot {(3^{4})}^{3}}$

Step 2.4

Multiply the exponents in ${(3^{4})}^{3}$ .

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Step 2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3^{- 22} \cdot 3^{4 \cdot 3}}$

Step 2.4.2

Multiply $4$ by $3$ .

$\frac{1}{3^{- 22} \cdot 3^{12}}$

Step 2.5

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

$\frac{1}{3^{- 22 + 12}}$

Step 2.6

Add $- 22$ and $12$ .

$\frac{1}{3^{- 10}}$

Step 3

Move $3^{- 10}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$3^{10}$

Step 4

Raise $3$ to the power of $10$ .

$59049$