Basic Math Examples

${(\frac{- 1}{10})}^{3} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

${(- \frac{1}{10})}^{3} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.2

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

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Step 1.2.1

Apply the product rule to $- \frac{1}{10}$ .

${(- 1)}^{3} {(\frac{1}{10})}^{3} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.2.2

Apply the product rule to $\frac{1}{10}$ .

${(- 1)}^{3} \frac{1^{3}}{10^{3}} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.3

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

$- \frac{1^{3}}{10^{3}} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.4

One to any power is one.

$- \frac{1}{10^{3}} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.5

Raise $10$ to the power of $3$ .

$- \frac{1}{1000} - {(\frac{- 1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.6

Move the negative in front of the fraction.

$- \frac{1}{1000} - {(- \frac{1}{10})}^{2} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.7

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

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Step 1.7.1

Apply the product rule to $- \frac{1}{10}$ .

$- \frac{1}{1000} - ({(- 1)}^{2} {(\frac{1}{10})}^{2}) - 2 \cdot \frac{- 1}{10} + 1$

Step 1.7.2

Apply the product rule to $\frac{1}{10}$ .

$- \frac{1}{1000} - ({(- 1)}^{2} \frac{1^{2}}{10^{2}}) - 2 \cdot \frac{- 1}{10} + 1$

Step 1.8

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

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Step 1.8.1

Move ${(- 1)}^{2}$ .

$- \frac{1}{1000} + {(- 1)}^{2} \cdot - 1 \frac{1^{2}}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.8.2

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

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Step 1.8.2.1

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

$- \frac{1}{1000} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{1^{2}}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.8.2.2

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

$- \frac{1}{1000} + {(- 1)}^{2 + 1} \frac{1^{2}}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.8.3

Add $2$ and $1$ .

$- \frac{1}{1000} + {(- 1)}^{3} \frac{1^{2}}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.9

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

$- \frac{1}{1000} - \frac{1^{2}}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.10

One to any power is one.

$- \frac{1}{1000} - \frac{1}{10^{2}} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.11

Raise $10$ to the power of $2$ .

$- \frac{1}{1000} - \frac{1}{100} - 2 \cdot \frac{- 1}{10} + 1$

Step 1.12

Cancel the common factor of $2$ .

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Step 1.12.1

Factor $2$ out of $- 2$ .

$- \frac{1}{1000} - \frac{1}{100} + 2 (- 1) \cdot \frac{- 1}{10} + 1$

Step 1.12.2

Factor $2$ out of $10$ .

$- \frac{1}{1000} - \frac{1}{100} + 2 \cdot - 1 \cdot \frac{- 1}{2 \cdot 5} + 1$

Step 1.12.3

Cancel the common factor.

$- \frac{1}{1000} - \frac{1}{100} + 2 \cdot - 1 \cdot \frac{- 1}{2 \cdot 5} + 1$

Step 1.12.4

Rewrite the expression.

$- \frac{1}{1000} - \frac{1}{100} - 1 \cdot \frac{- 1}{5} + 1$

Step 1.13

Move the negative in front of the fraction.

$- \frac{1}{1000} - \frac{1}{100} - 1 \cdot (- \frac{1}{5}) + 1$

Step 1.14

Multiply $- 1 (- \frac{1}{5})$ .

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Step 1.14.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{1000} - \frac{1}{100} + 1 (\frac{1}{5}) + 1$

Step 1.14.2

Multiply $\frac{1}{5}$ by $1$ .

$- \frac{1}{1000} - \frac{1}{100} + \frac{1}{5} + 1$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{1}{100}$ by $\frac{10}{10}$ .

$- \frac{1}{1000} - (\frac{1}{100} \cdot \frac{10}{10}) + \frac{1}{5} + 1$

Step 2.2

Multiply $\frac{1}{100}$ by $\frac{10}{10}$ .

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{1}{5} + 1$

Step 2.3

Multiply $\frac{1}{5}$ by $\frac{200}{200}$ .

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{1}{5} \cdot \frac{200}{200} + 1$

Step 2.4

Multiply $\frac{1}{5}$ by $\frac{200}{200}$ .

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{200}{5 \cdot 200} + 1$

Step 2.5

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

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{200}{5 \cdot 200} + \frac{1}{1}$

Step 2.6

Multiply $\frac{1}{1}$ by $\frac{1000}{1000}$ .

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{200}{5 \cdot 200} + \frac{1}{1} \cdot \frac{1000}{1000}$

Step 2.7

Multiply $\frac{1}{1}$ by $\frac{1000}{1000}$ .

$- \frac{1}{1000} - \frac{10}{100 \cdot 10} + \frac{200}{5 \cdot 200} + \frac{1000}{1000}$

Step 2.8

Reorder the factors of $100 \cdot 10$ .

$- \frac{1}{1000} - \frac{10}{10 \cdot 100} + \frac{200}{5 \cdot 200} + \frac{1000}{1000}$

Step 2.9

Multiply $10$ by $100$ .

$- \frac{1}{1000} - \frac{10}{1000} + \frac{200}{5 \cdot 200} + \frac{1000}{1000}$

Step 2.10

Multiply $5$ by $200$ .

$- \frac{1}{1000} - \frac{10}{1000} + \frac{200}{1000} + \frac{1000}{1000}$

Step 3

Combine the numerators over the common denominator.

$\frac{- 1 - 10 + 200 + 1000}{1000}$

Step 4

Simplify by adding and subtracting.

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Step 4.1

Subtract $10$ from $- 1$ .

$\frac{- 11 + 200 + 1000}{1000}$

Step 4.2

Add $- 11$ and $200$ .

$\frac{189 + 1000}{1000}$

Step 4.3

Add $189$ and $1000$ .

$\frac{1189}{1000}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1189}{1000}$

Decimal Form:

$1.189$

Mixed Number Form:

$1 \frac{189}{1000}$