Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{1}{3}$ .

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

Step 1.1.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 1.2

Multiply ${(- 1)}^{2}$ by $- 1$ by adding the exponents.

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Step 1.2.1

Move $- 1$ .

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

Step 1.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.3

Add $1$ and $2$ .

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

Step 1.3

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

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

Step 1.4

One to any power is one.

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

Step 1.5

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

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

Step 1.6

Multiply $- \frac{1}{9} \cdot \frac{1}{3}$ .

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Step 1.6.1

Multiply $\frac{1}{3}$ by $\frac{1}{9}$ .

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

Step 1.6.2

Multiply $3$ by $9$ .

$- \frac{1}{27} - 3 - 3^{2}$

Step 1.7

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

$- \frac{1}{27} - 3 - 1 \cdot 9$

Step 1.8

Multiply $- 1$ by $9$ .

$- \frac{1}{27} - 3 - 9$

Step 2

Find the common denominator.

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Step 2.1

Write $- 3$ as a fraction with denominator $1$ .

$- \frac{1}{27} + \frac{- 3}{1} - 9$

Step 2.2

Multiply $\frac{- 3}{1}$ by $\frac{27}{27}$ .

$- \frac{1}{27} + \frac{- 3}{1} \cdot \frac{27}{27} - 9$

Step 2.3

Multiply $\frac{- 3}{1}$ by $\frac{27}{27}$ .

$- \frac{1}{27} + \frac{- 3 \cdot 27}{27} - 9$

Step 2.4

Write $- 9$ as a fraction with denominator $1$ .

$- \frac{1}{27} + \frac{- 3 \cdot 27}{27} + \frac{- 9}{1}$

Step 2.5

Multiply $\frac{- 9}{1}$ by $\frac{27}{27}$ .

$- \frac{1}{27} + \frac{- 3 \cdot 27}{27} + \frac{- 9}{1} \cdot \frac{27}{27}$

Step 2.6

Multiply $\frac{- 9}{1}$ by $\frac{27}{27}$ .

$- \frac{1}{27} + \frac{- 3 \cdot 27}{27} + \frac{- 9 \cdot 27}{27}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 1 - 3 \cdot 27 - 9 \cdot 27}{27}$

Step 4

Simplify each term.

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Step 4.1

Multiply $- 3$ by $27$ .

$\frac{- 1 - 81 - 9 \cdot 27}{27}$

Step 4.2

Multiply $- 9$ by $27$ .

$\frac{- 1 - 81 - 243}{27}$

Step 5

Simplify the expression.

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Step 5.1

Subtract $81$ from $- 1$ .

$\frac{- 82 - 243}{27}$

Step 5.2

Subtract $243$ from $- 82$ .

$\frac{- 325}{27}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{325}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{325}{27}$

Decimal Form:

$- 12.\overline{037}$

Mixed Number Form:

$- 12 \frac{1}{27}$