Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Multiply $5^{3}$ by $5^{- 2}$ by adding the exponents.

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Step 1.1.1

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

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

Step 1.1.2

Subtract $2$ from $3$ .

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

Step 1.2

Simplify $5^{1}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Combine $5$ and $\frac{1}{9}$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

${(\frac{2}{3})}^{2} \frac{\frac{1}{9} \cdot - 27}{{(3^{- 1})}^{- 3}} \cdot \frac{5}{\frac{5}{9}}$

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

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

${(\frac{2}{3})}^{2} \frac{\frac{1}{9} \cdot - 27}{3^{- - 3}} \cdot \frac{5}{\frac{5}{9}}$

Step 5.1.2

Multiply $- 1$ by $- 3$ .

${(\frac{2}{3})}^{2} \frac{\frac{1}{9} \cdot - 27}{3^{3}} \cdot \frac{5}{\frac{5}{9}}$

Step 5.2

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

${(\frac{2}{3})}^{2} \frac{\frac{1}{9} \cdot - 27}{27} \cdot \frac{5}{\frac{5}{9}}$

Step 6

Simplify terms.

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Step 6.1

Combine $\frac{1}{9}$ and $- 27$ .

${(\frac{2}{3})}^{2} \frac{\frac{- 27}{9}}{27} \cdot \frac{5}{\frac{5}{9}}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Divide $- 27$ by $9$ .

${(\frac{2}{3})}^{2} \frac{- 3}{27} \cdot \frac{5}{\frac{5}{9}}$

Step 6.2.2

Apply the product rule to $\frac{2}{3}$ .

$\frac{2^{2}}{3^{2}} \cdot \frac{- 3}{27} \cdot \frac{5}{\frac{5}{9}}$

Step 6.3

Combine.

$\frac{2^{2} \cdot - 3}{3^{2} \cdot 27} \cdot \frac{5}{\frac{5}{9}}$

Step 6.4

Combine.

$\frac{2^{2} \cdot - 3 \cdot 5}{3^{2} \cdot 27 \frac{5}{9}}$

Step 6.5

Cancel the common factor of $- 3$ and $27$ .

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Step 6.5.1

Factor $3$ out of $2^{2} \cdot - 3 \cdot 5$ .

$\frac{3 (2^{2} \cdot - 1 \cdot 5)}{3^{2} \cdot 27 \frac{5}{9}}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor $3$ out of $3^{2} \cdot 27 \frac{5}{9}$ .

$\frac{3 (2^{2} \cdot - 1 \cdot 5)}{3 (3 \cdot 27 \frac{5}{9})}$

Step 6.5.2.2

Cancel the common factor.

$\frac{3 (2^{2} \cdot - 1 \cdot 5)}{3 (3 \cdot 27 \frac{5}{9})}$

Step 6.5.2.3

Rewrite the expression.

$\frac{2^{2} \cdot - 1 \cdot 5}{3 \cdot 27 \frac{5}{9}}$

Step 6.6

Multiply $5$ by $- 1$ .

$\frac{2^{2} \cdot - 5}{3 \cdot 27 \frac{5}{9}}$

Step 7

Simplify the denominator.

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Step 7.1

Multiply $3$ by $27$ .

$\frac{2^{2} \cdot - 5}{81 (\frac{5}{9})}$

Step 7.2

Combine $81$ and $\frac{5}{9}$ .

$\frac{2^{2} \cdot - 5}{\frac{81 \cdot 5}{9}}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Multiply $81$ by $5$ .

$\frac{2^{2} \cdot - 5}{\frac{405}{9}}$

Step 8.2

Simplify the expression.

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Step 8.2.1

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

$\frac{4 \cdot - 5}{\frac{405}{9}}$

Step 8.2.2

Divide $405$ by $9$ .

$\frac{4 \cdot - 5}{45}$

Step 8.2.3

Multiply $4$ by $- 5$ .

$\frac{- 20}{45}$

Step 8.3

Cancel the common factor of $- 20$ and $45$ .

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Step 8.3.1

Factor $5$ out of $- 20$ .

$\frac{5 (- 4)}{45}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $5$ out of $45$ .

$\frac{5 \cdot - 4}{5 \cdot 9}$

Step 8.3.2.2

Cancel the common factor.

$\frac{5 \cdot - 4}{5 \cdot 9}$

Step 8.3.2.3

Rewrite the expression.

$\frac{- 4}{9}$

Step 8.4

Move the negative in front of the fraction.

$- \frac{4}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{9}$

Decimal Form:

$- 0.\overline{4}$