Basic Math Examples

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{2^{4}}{3^{5}} \cdot \frac{2^{- 3}}{3^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{2^{4}}{3^{5}} \cdot \frac{\frac{1}{2^{3}}}{3^{2}}$

Step 1.2

Raise $2$ to the power of $3$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{2^{4}}{3^{5}} \cdot \frac{\frac{1}{8}}{3^{2}}$

Step 2

Evaluate the exponents.

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Step 2.1

Raise $3$ to the power of $2$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{2^{4}}{3^{5}} \cdot \frac{\frac{1}{8}}{9}$

Step 2.2

Raise $2$ to the power of $4$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{16}{3^{5}} \cdot \frac{\frac{1}{8}}{9}$

Step 2.3

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

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{3^{4}}{2^{- 5}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 2.4

Raise $3$ to the power of $4$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{81}{2^{- 5}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{81}{\frac{1}{2^{5}}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 3.2

Raise $2$ to the power of $5$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{81}{\frac{1}{32}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $2^{- 2}$ .

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Step 4.1.1

Cancel the common factor.

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{2^{- 2}}{2^{- 2}} \cdot \frac{81}{\frac{1}{32}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 4.1.2

Rewrite the expression.

$\frac{3^{- 2}}{3^{- 3}} \cdot 1 \cdot \frac{81}{\frac{1}{32}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 4.2

Multiply $\frac{3^{- 2}}{3^{- 3}}$ by $1$ .

$\frac{3^{- 2}}{3^{- 3}} \cdot \frac{81}{\frac{1}{32}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 4.3

Combine.

$\frac{3^{- 2} \cdot 81}{3^{- 3} \frac{1}{32}} \cdot \frac{16}{243} \cdot \frac{\frac{1}{8}}{9}$

Step 4.4

Combine.

$\frac{3^{- 2} \cdot 81 \cdot 16}{3^{- 3} \frac{1}{32} \cdot 243} \cdot \frac{\frac{1}{8}}{9}$

Step 4.5

Combine.

$\frac{3^{- 2} \cdot 81 \cdot 16 \frac{1}{8}}{3^{- 3} \frac{1}{32} \cdot 243 \cdot 9}$

Step 4.6

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

$\frac{3^{- 2} \cdot 81 \cdot 16 \frac{1}{8} \cdot 3^{3}}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $3^{- 2}$ by $3^{3}$ by adding the exponents.

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Step 5.1.1

Move $3^{3}$ .

$\frac{3^{3} \cdot 3^{- 2} \cdot 81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 5.1.2

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

$\frac{3^{3 - 2} \cdot 81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 5.1.3

Subtract $2$ from $3$ .

$\frac{3^{1} \cdot 81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 5.2

Simplify $3^{1} \cdot 81 \cdot 16 \frac{1}{8}$ .

$\frac{3 \cdot 81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 6

Cancel the common factor of $3$ and $243$ .

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Step 6.1

Factor $3$ out of $3 \cdot 81 \cdot 16 \frac{1}{8}$ .

$\frac{3 (81 \cdot 16 \frac{1}{8})}{\frac{1}{32} \cdot 243 \cdot 9}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $3$ out of $\frac{1}{32} \cdot 243 \cdot 9$ .

$\frac{3 (81 \cdot 16 \frac{1}{8})}{3 (\frac{1}{32} \cdot 81 \cdot 9)}$

Step 6.2.2

Cancel the common factor.

$\frac{3 (81 \cdot 16 \frac{1}{8})}{3 (\frac{1}{32} \cdot 81 \cdot 9)}$

Step 6.2.3

Rewrite the expression.

$\frac{81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 81 \cdot 9}$

Step 7

Cancel the common factor of $81$ .

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Step 7.1

Cancel the common factor.

$\frac{81 \cdot 16 \frac{1}{8}}{\frac{1}{32} \cdot 81 \cdot 9}$

Step 7.2

Rewrite the expression.

$\frac{16 (\frac{1}{8})}{\frac{1}{32} \cdot 9}$

Step 8

Combine $16$ and $\frac{1}{8}$ .

$\frac{\frac{16}{8}}{\frac{1}{32} \cdot 9}$

Step 9

Combine $\frac{1}{32}$ and $9$ .

$\frac{\frac{16}{8}}{\frac{9}{32}}$

Step 10

Divide $16$ by $8$ .

$\frac{2}{\frac{9}{32}}$

Step 11

Multiply the numerator by the reciprocal of the denominator.

$2 (\frac{32}{9})$

Step 12

Multiply $2 (\frac{32}{9})$ .

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Step 12.1

Combine $2$ and $\frac{32}{9}$ .

$\frac{2 \cdot 32}{9}$

Step 12.2

Multiply $2$ by $32$ .

$\frac{64}{9}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{64}{9}$

Decimal Form:

$7.\overline{1}$

Mixed Number Form:

$7 \frac{1}{9}$