Algebra Examples

$\frac{{(- 3^{2})}^{3} \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $- 3^{2}$ .

$\frac{{(- 1)}^{3} {(3^{2})}^{3} \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.2

Raise $- 1$ to the power of $3$ .

$\frac{- {(3^{2})}^{3} \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.3

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

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Step 1.3.1

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

$\frac{- 3^{2 \cdot 3} \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.3.2

Multiply $2$ by $3$ .

$\frac{- 3^{6} \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.4

Raise $3$ to the power of $6$ .

$\frac{- 1 \cdot 729 \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.5

Combine exponents.

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Step 1.5.1

Multiply $- 1$ by $729$ .

$\frac{- 729 \cdot 27}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 1.5.2

Multiply $- 729$ by $27$ .

$\frac{- 19683}{9^{4} \cdot {(\frac{1}{3})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{- 19683}{{(3^{2})}^{4} {(\frac{1}{3})}^{3}}$

Step 2.2

Multiply the exponents in ${(3^{2})}^{4}$ .

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Step 2.2.1

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

$\frac{- 19683}{3^{2 \cdot 4} {(\frac{1}{3})}^{3}}$

Step 2.2.2

Multiply $2$ by $4$ .

$\frac{- 19683}{3^{8} {(\frac{1}{3})}^{3}}$

Step 2.3

Rewrite $\frac{1}{3}$ as $3^{- 1}$ .

$\frac{- 19683}{3^{8} {(3^{- 1})}^{3}}$

Step 2.4

Raise $3$ to the power of $1$ .

$\frac{- 19683}{3^{8} {({(3^{1})}^{- 1})}^{3}}$

Step 2.5

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

$\frac{- 19683}{3^{8} {(3^{1 \cdot - 1})}^{3}}$

Step 2.6

Multiply $- 1$ by $1$ .

$\frac{- 19683}{3^{8} {(3^{- 1})}^{3}}$

Step 2.7

Multiply the exponents in ${(3^{- 1})}^{3}$ .

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Step 2.7.1

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

$\frac{- 19683}{3^{8} \cdot 3^{- 1 \cdot 3}}$

Step 2.7.2

Multiply $- 1$ by $3$ .

$\frac{- 19683}{3^{8} \cdot 3^{- 3}}$

Step 2.8

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

$\frac{- 19683}{3^{8 - 3}}$

Step 2.9

Subtract $3$ from $8$ .

$\frac{- 19683}{3^{5}}$

Step 3

Simplify the expression.

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Step 3.1

Raise $3$ to the power of $5$ .

$\frac{- 19683}{243}$

Step 3.2

Divide $- 19683$ by $243$ .

$- 81$