Calculus Examples

$2 {(- \frac{1}{2})}^{3} + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

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 $- \frac{1}{2}$ .

$2 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.1.2

Apply the product rule to $\frac{1}{2}$ .

$2 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.2

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

$2 (- \frac{1^{3}}{2^{3}}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.3

One to any power is one.

$2 (- \frac{1}{2^{3}}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.4

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

$2 (- \frac{1}{8}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$2 (\frac{- 1}{8}) + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.5.2

Factor $2$ out of $8$ .

$2 \frac{- 1}{2 (4)} + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.5.3

Cancel the common factor.

$2 \frac{- 1}{2 \cdot 4} + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.5.4

Rewrite the expression.

$\frac{- 1}{4} + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

Step 1.6

Move the negative in front of the fraction.

$- \frac{1}{4} + 3 {(- \frac{1}{2})}^{2} - 72 (- \frac{1}{2})$

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 $- \frac{1}{2}$ .

$- \frac{1}{4} + 3 ({(- 1)}^{2} {(\frac{1}{2})}^{2}) - 72 (- \frac{1}{2})$

Step 1.7.2

Apply the product rule to $\frac{1}{2}$ .

$- \frac{1}{4} + 3 ({(- 1)}^{2} \frac{1^{2}}{2^{2}}) - 72 (- \frac{1}{2})$

Step 1.8

Raise $- 1$ to the power of $2$ .

$- \frac{1}{4} + 3 (1 \frac{1^{2}}{2^{2}}) - 72 (- \frac{1}{2})$

Step 1.9

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

$- \frac{1}{4} + 3 \frac{1^{2}}{2^{2}} - 72 (- \frac{1}{2})$

Step 1.10

One to any power is one.

$- \frac{1}{4} + 3 \frac{1}{2^{2}} - 72 (- \frac{1}{2})$

Step 1.11

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

$- \frac{1}{4} + 3 (\frac{1}{4}) - 72 (- \frac{1}{2})$

Step 1.12

Combine $3$ and $\frac{1}{4}$ .

$- \frac{1}{4} + \frac{3}{4} - 72 (- \frac{1}{2})$

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{1}{4} + \frac{3}{4} - 72 (\frac{- 1}{2})$

Step 1.13.2

Factor $2$ out of $- 72$ .

$- \frac{1}{4} + \frac{3}{4} + 2 (- 36) \frac{- 1}{2}$

Step 1.13.3

Cancel the common factor.

$- \frac{1}{4} + \frac{3}{4} + 2 \cdot - 36 \frac{- 1}{2}$

Step 1.13.4

Rewrite the expression.

$- \frac{1}{4} + \frac{3}{4} - 36 \cdot - 1$

Step 1.14

Multiply $- 36$ by $- 1$ .

$- \frac{1}{4} + \frac{3}{4} + 36$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$36 + \frac{- 1 + 3}{4}$

Step 2.2

Add $- 1$ and $3$ .

$36 + \frac{2}{4}$

Step 2.3

Cancel the common factor of $2$ and $4$ .

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Step 2.3.1

Factor $2$ out of $2$ .

$36 + \frac{2 (1)}{4}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

$36 + \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.3.2.2

Cancel the common factor.

$36 + \frac{2 \cdot 1}{2 \cdot 2}$

Step 2.3.2.3

Rewrite the expression.

$36 + \frac{1}{2}$

Step 3

To write $36$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$36 \cdot \frac{2}{2} + \frac{1}{2}$

Step 4

Combine $36$ and $\frac{2}{2}$ .

$\frac{36 \cdot 2}{2} + \frac{1}{2}$

Step 5

Combine the numerators over the common denominator.

$\frac{36 \cdot 2 + 1}{2}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $36$ by $2$ .

$\frac{72 + 1}{2}$

Step 6.2

Add $72$ and $1$ .

$\frac{73}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{73}{2}$

Decimal Form:

$36.5$

Mixed Number Form:

$36 \frac{1}{2}$