Calculus Examples

$R (2000) = 15 {(2 (2000) + 1)}^{- 1} + 30 (2000) - 15$

Step 1

Simplify.

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Step 1.1

Multiply $2$ by $2000$ .

$R (2000) = 15 {(4000 + 1)}^{- 1} + 30 (2000) - 15$

Step 1.2

Add $4000$ and $1$ .

$R (2000) = 15 \cdot 4001^{- 1} + 30 (2000) - 15$

Step 1.3

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

$R (2000) = 15 \cdot \frac{1}{4001} + 30 (2000) - 15$

Step 1.4

Combine $15$ and $\frac{1}{4001}$ .

$R (2000) = \frac{15}{4001} + 30 (2000) - 15$

Step 1.5

Multiply $30$ by $2000$ .

$R (2000) = \frac{15}{4001} + 60000 - 15$

Step 1.6

To write $60000$ as a fraction with a common denominator, multiply by $\frac{4001}{4001}$ .

$R (2000) = \frac{15}{4001} + 60000 \cdot \frac{4001}{4001} - 15$

Step 1.7

Combine $60000$ and $\frac{4001}{4001}$ .

$R (2000) = \frac{15}{4001} + \frac{60000 \cdot 4001}{4001} - 15$

Step 1.8

Combine the numerators over the common denominator.

$R (2000) = \frac{15 + 60000 \cdot 4001}{4001} - 15$

Step 1.9

Simplify the numerator.

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Step 1.9.1

Multiply $60000$ by $4001$ .

$R (2000) = \frac{15 + 240060000}{4001} - 15$

Step 1.9.2

Add $15$ and $240060000$ .

$R (2000) = \frac{240060015}{4001} - 15$

Step 1.10

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{4001}{4001}$ .

$R (2000) = \frac{240060015}{4001} - 15 \cdot \frac{4001}{4001}$

Step 1.11

Combine $- 15$ and $\frac{4001}{4001}$ .

$R (2000) = \frac{240060015}{4001} + \frac{- 15 \cdot 4001}{4001}$

Step 1.12

Combine the numerators over the common denominator.

$R (2000) = \frac{240060015 - 15 \cdot 4001}{4001}$

Step 1.13

Simplify the numerator.

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Step 1.13.1

Multiply $- 15$ by $4001$ .

$R (2000) = \frac{240060015 - 60015}{4001}$

Step 1.13.2

Subtract $60015$ from $240060015$ .

$R (2000) = \frac{240000000}{4001}$

Step 2

Divide each term in $R (2000) = \frac{240000000}{4001}$ by $2000$ and simplify.

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Step 2.1

Divide each term in $R (2000) = \frac{240000000}{4001}$ by $2000$ .

$\frac{R (2000)}{2000} = \frac{\frac{240000000}{4001}}{2000}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2000$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{R \cdot 2000}{2000} = \frac{\frac{240000000}{4001}}{2000}$

Step 2.2.1.2

Divide $R$ by $1$ .

$R = \frac{\frac{240000000}{4001}}{2000}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$R = \frac{240000000}{4001} \cdot \frac{1}{2000}$

Step 2.3.2

Cancel the common factor of $2000$ .

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Step 2.3.2.1

Factor $2000$ out of $240000000$ .

$R = \frac{2000 (120000)}{4001} \cdot \frac{1}{2000}$

Step 2.3.2.2

Cancel the common factor.

$R = \frac{2000 \cdot 120000}{4001} \cdot \frac{1}{2000}$

Step 2.3.2.3

Rewrite the expression.

$R = \frac{120000}{4001}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$R = \frac{120000}{4001}$

Decimal Form:

$R = 29.99250187 \dots$

Mixed Number Form:

$R = 29 \frac{3971}{4001}$