Basic Math Examples

${(- \frac{1}{3})}^{4} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

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 $- \frac{1}{3}$ .

${(- 1)}^{4} {(\frac{1}{3})}^{4} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.1.2

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

${(- 1)}^{4} \frac{1^{4}}{3^{4}} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.2

Raise $- 1$ to the power of $4$ .

$1 \frac{1^{4}}{3^{4}} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.3

Multiply $\frac{1^{4}}{3^{4}}$ by $1$ .

$\frac{1^{4}}{3^{4}} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.4

One to any power is one.

$\frac{1}{3^{4}} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.5

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

$\frac{1}{81} - {(\frac{1}{6})}^{2} + {(- \frac{1}{3})}^{3}$

Step 1.6

Apply the product rule to $\frac{1}{6}$ .

$\frac{1}{81} - \frac{1^{2}}{6^{2}} + {(- \frac{1}{3})}^{3}$

Step 1.7

One to any power is one.

$\frac{1}{81} - \frac{1}{6^{2}} + {(- \frac{1}{3})}^{3}$

Step 1.8

Raise $6$ to the power of $2$ .

$\frac{1}{81} - \frac{1}{36} + {(- \frac{1}{3})}^{3}$

Step 1.9

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

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Step 1.9.1

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

$\frac{1}{81} - \frac{1}{36} + {(- 1)}^{3} {(\frac{1}{3})}^{3}$

Step 1.9.2

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

$\frac{1}{81} - \frac{1}{36} + {(- 1)}^{3} \frac{1^{3}}{3^{3}}$

Step 1.10

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

$\frac{1}{81} - \frac{1}{36} - \frac{1^{3}}{3^{3}}$

Step 1.11

One to any power is one.

$\frac{1}{81} - \frac{1}{36} - \frac{1}{3^{3}}$

Step 1.12

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

$\frac{1}{81} - \frac{1}{36} - \frac{1}{27}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{1}{81}$ by $\frac{4}{4}$ .

$\frac{1}{81} \cdot \frac{4}{4} - \frac{1}{36} - \frac{1}{27}$

Step 2.2

Multiply $\frac{1}{81}$ by $\frac{4}{4}$ .

$\frac{4}{81 \cdot 4} - \frac{1}{36} - \frac{1}{27}$

Step 2.3

Multiply $\frac{1}{36}$ by $\frac{9}{9}$ .

$\frac{4}{81 \cdot 4} - (\frac{1}{36} \cdot \frac{9}{9}) - \frac{1}{27}$

Step 2.4

Multiply $\frac{1}{36}$ by $\frac{9}{9}$ .

$\frac{4}{81 \cdot 4} - \frac{9}{36 \cdot 9} - \frac{1}{27}$

Step 2.5

Multiply $\frac{1}{27}$ by $\frac{12}{12}$ .

$\frac{4}{81 \cdot 4} - \frac{9}{36 \cdot 9} - (\frac{1}{27} \cdot \frac{12}{12})$

Step 2.6

Multiply $\frac{1}{27}$ by $\frac{12}{12}$ .

$\frac{4}{81 \cdot 4} - \frac{9}{36 \cdot 9} - \frac{12}{27 \cdot 12}$

Step 2.7

Reorder the factors of $81 \cdot 4$ .

$\frac{4}{4 \cdot 81} - \frac{9}{36 \cdot 9} - \frac{12}{27 \cdot 12}$

Step 2.8

Multiply $4$ by $81$ .

$\frac{4}{324} - \frac{9}{36 \cdot 9} - \frac{12}{27 \cdot 12}$

Step 2.9

Reorder the factors of $36 \cdot 9$ .

$\frac{4}{324} - \frac{9}{9 \cdot 36} - \frac{12}{27 \cdot 12}$

Step 2.10

Multiply $9$ by $36$ .

$\frac{4}{324} - \frac{9}{324} - \frac{12}{27 \cdot 12}$

Step 2.11

Reorder the factors of $27 \cdot 12$ .

$\frac{4}{324} - \frac{9}{324} - \frac{12}{12 \cdot 27}$

Step 2.12

Multiply $12$ by $27$ .

$\frac{4}{324} - \frac{9}{324} - \frac{12}{324}$

Step 3

Combine the numerators over the common denominator.

$\frac{4 - 9 - 12}{324}$

Step 4

Simplify the expression.

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Step 4.1

Subtract $9$ from $4$ .

$\frac{- 5 - 12}{324}$

Step 4.2

Subtract $12$ from $- 5$ .

$\frac{- 17}{324}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{17}{324}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{17}{324}$

Decimal Form:

$- 0.05246913 \dots$