Calculus Examples

$f (x) = 0.004 x^{3} - 2 x^{2} + x + 3$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $0.004 x^{3} - 2 x^{2} + x + 3$ with respect to $x$ is $\frac{d}{d x} [0.004 x^{3}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [0.004 x^{3}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.2

Evaluate $\frac{d}{d x} [0.004 x^{3}]$ .

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Step 1.1.2.1

Since $0.004$ is constant with respect to $x$ , the derivative of $0.004 x^{3}$ with respect to $x$ is $0.004 \frac{d}{d x} [x^{3}]$ .

$0.004 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0.004 (3 x^{2}) + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.2.3

Multiply $3$ by $0.004$ .

$0.012 x^{2} + \frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 2 x^{2}]$ .

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Step 1.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{2}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{2}]$ .

$0.012 x^{2} - 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0.012 x^{2} - 2 (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.3.3

Multiply $2$ by $- 2$ .

$0.012 x^{2} - 4 x + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0.012 x^{2} - 4 x + 1 + \frac{d}{d x} [3]$

Step 1.1.4.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0.012 x^{2} - 4 x + 1 + 0$

Step 1.1.4.3

Add $0.012 x^{2} - 4 x + 1$ and $0$ .

$f' (x) = 0.012 x^{2} - 4 x + 1$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $0.012 x^{2} - 4 x + 1$ .

$0.012 x^{2} - 4 x + 1$

Step 2

Set the first derivative equal to $0$ then solve the equation $0.012 x^{2} - 4 x + 1 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$0.012 x^{2} - 4 x + 1 = 0$

Step 2.2

Factor $0.004$ out of $0.012 x^{2} - 4 x + 1$ .

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Step 2.2.1

Factor $0.004$ out of $0.012 x^{2}$ .

$0.004 (3 x^{2}) - 4 x + 1 = 0$

Step 2.2.2

Factor $0.004$ out of $- 4 x$ .

$0.004 (3 x^{2}) + 0.004 (- 1000 x) + 1 = 0$

Step 2.2.3

Factor $0.004$ out of $1$ .

$0.004 (3 x^{2}) + 0.004 (- 1000 x) + 0.004 (250) = 0$

Step 2.2.4

Factor $0.004$ out of $0.004 (3 x^{2}) + 0.004 (- 1000 x)$ .

$0.004 (3 x^{2} - 1000 x) + 0.004 (250) = 0$

Step 2.2.5

Factor $0.004$ out of $0.004 (3 x^{2} - 1000 x) + 0.004 (250)$ .

$0.004 (3 x^{2} - 1000 x + 250) = 0$

Step 2.3

Divide each term in $0.004 (3 x^{2} - 1000 x + 250) = 0$ by $0.004$ and simplify.

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Step 2.3.1

Divide each term in $0.004 (3 x^{2} - 1000 x + 250) = 0$ by $0.004$ .

$\frac{0.004 (3 x^{2} - 1000 x + 250)}{0.004} = \frac{0}{0.004}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $0.004$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{0.004 (3 x^{2} - 1000 x + 250)}{0.004} = \frac{0}{0.004}$

Step 2.3.2.1.2

Divide $3 x^{2} - 1000 x + 250$ by $1$ .

$3 x^{2} - 1000 x + 250 = \frac{0}{0.004}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $0.004$ .

$3 x^{2} - 1000 x + 250 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5

Substitute the values $a = 3$ , $b = - 1000$ , and $c = 250$ into the quadratic formula and solve for $x$ .

$\frac{1000 \pm \sqrt{{(- 1000)}^{2} - 4 \cdot (3 \cdot 250)}}{2 \cdot 3}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 1000$ to the power of $2$ .

$x = \frac{1000 \pm \sqrt{1000000 - 4 \cdot 3 \cdot 250}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot 250$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{1000 \pm \sqrt{1000000 - 12 \cdot 250}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $250$ .

$x = \frac{1000 \pm \sqrt{1000000 - 3000}}{2 \cdot 3}$

Step 2.6.1.3

Subtract $3000$ from $1000000$ .

$x = \frac{1000 \pm \sqrt{997000}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $997000$ as $10^{2} \cdot 9970$ .

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Step 2.6.1.4.1

Factor $100$ out of $997000$ .

$x = \frac{1000 \pm \sqrt{100 (9970)}}{2 \cdot 3}$

Step 2.6.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{1000 \pm \sqrt{10^{2} \cdot 9970}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{1000 \pm 10 \sqrt{9970}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{1000 \pm 10 \sqrt{9970}}{6}$

Step 2.6.3

Simplify $\frac{1000 \pm 10 \sqrt{9970}}{6}$ .

$x = \frac{500 \pm 5 \sqrt{9970}}{3}$

Step 2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 1000$ to the power of $2$ .

$x = \frac{1000 \pm \sqrt{1000000 - 4 \cdot 3 \cdot 250}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot 250$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{1000 \pm \sqrt{1000000 - 12 \cdot 250}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $250$ .

$x = \frac{1000 \pm \sqrt{1000000 - 3000}}{2 \cdot 3}$

Step 2.7.1.3

Subtract $3000$ from $1000000$ .

$x = \frac{1000 \pm \sqrt{997000}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $997000$ as $10^{2} \cdot 9970$ .

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Step 2.7.1.4.1

Factor $100$ out of $997000$ .

$x = \frac{1000 \pm \sqrt{100 (9970)}}{2 \cdot 3}$

Step 2.7.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{1000 \pm \sqrt{10^{2} \cdot 9970}}{2 \cdot 3}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{1000 \pm 10 \sqrt{9970}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{1000 \pm 10 \sqrt{9970}}{6}$

Step 2.7.3

Simplify $\frac{1000 \pm 10 \sqrt{9970}}{6}$ .

$x = \frac{500 \pm 5 \sqrt{9970}}{3}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = \frac{500 + 5 \sqrt{9970}}{3}$

Step 2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $- 1000$ to the power of $2$ .

$x = \frac{1000 \pm \sqrt{1000000 - 4 \cdot 3 \cdot 250}}{2 \cdot 3}$

Step 2.8.1.2

Multiply $- 4 \cdot 3 \cdot 250$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{1000 \pm \sqrt{1000000 - 12 \cdot 250}}{2 \cdot 3}$

Step 2.8.1.2.2

Multiply $- 12$ by $250$ .

$x = \frac{1000 \pm \sqrt{1000000 - 3000}}{2 \cdot 3}$

Step 2.8.1.3

Subtract $3000$ from $1000000$ .

$x = \frac{1000 \pm \sqrt{997000}}{2 \cdot 3}$

Step 2.8.1.4

Rewrite $997000$ as $10^{2} \cdot 9970$ .

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Step 2.8.1.4.1

Factor $100$ out of $997000$ .

$x = \frac{1000 \pm \sqrt{100 (9970)}}{2 \cdot 3}$

Step 2.8.1.4.2

Rewrite $100$ as $10^{2}$ .

$x = \frac{1000 \pm \sqrt{10^{2} \cdot 9970}}{2 \cdot 3}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{1000 \pm 10 \sqrt{9970}}{2 \cdot 3}$

Step 2.8.2

Multiply $2$ by $3$ .

$x = \frac{1000 \pm 10 \sqrt{9970}}{6}$

Step 2.8.3

Simplify $\frac{1000 \pm 10 \sqrt{9970}}{6}$ .

$x = \frac{500 \pm 5 \sqrt{9970}}{3}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = \frac{500 - 5 \sqrt{9970}}{3}$

Step 2.9

The final answer is the combination of both solutions.

$x = \frac{500 + 5 \sqrt{9970}}{3}, \frac{500 - 5 \sqrt{9970}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $0.004 x^{3} - 2 x^{2} + x + 3$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{500 + 5 \sqrt{9970}}{3}$ .

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Step 4.1.1

Substitute $\frac{500 + 5 \sqrt{9970}}{3}$ for $x$ .

$0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Remove parentheses.

$0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Write $0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3}$ as a fraction with denominator $1$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3}}{1} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.2

Multiply $\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3}}{1} \cdot \frac{3}{3} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.3

Multiply $\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.4

Write $- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2}$ as a fraction with denominator $1$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2}}{1} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.5

Multiply $\frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2}}{1} \cdot \frac{3}{3} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.6

Multiply $\frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 + 5 \sqrt{9970}}{3} + 3$

Step 4.1.2.2.7

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

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 + 5 \sqrt{9970}}{3} + \frac{3}{1}$

Step 4.1.2.2.8

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

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 + 5 \sqrt{9970}}{3} + \frac{3}{1} \cdot \frac{3}{3}$

Step 4.1.2.2.9

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

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 + 5 \sqrt{9970}}{3} + \frac{3 \cdot 3}{3}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{0.004 {(\frac{500 + 5 \sqrt{9970}}{3})}^{3} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Apply the product rule to $\frac{500 + 5 \sqrt{9970}}{3}$ .

$\frac{0.004 \frac{{(500 + 5 \sqrt{9970})}^{3}}{3^{3}} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.2

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

$\frac{0.004 \frac{{(500 + 5 \sqrt{9970})}^{3}}{27} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.3

Cancel the common factor of $0.004$ .

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Step 4.1.2.4.3.1

Factor $0.004$ out of $27$ .

$\frac{0.004 \frac{{(500 + 5 \sqrt{9970})}^{3}}{0.004 (6750)} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.3.2

Cancel the common factor.

$\frac{0.004 \frac{{(500 + 5 \sqrt{9970})}^{3}}{0.004 \cdot 6750} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.3.3

Rewrite the expression.

$\frac{\frac{{(500 + 5 \sqrt{9970})}^{3}}{6750} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.4

Cancel the common factor of $3$ .

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Step 4.1.2.4.4.1

Factor $3$ out of $6750$ .

$\frac{\frac{{(500 + 5 \sqrt{9970})}^{3}}{3 (2250)} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.4.2

Cancel the common factor.

$\frac{\frac{{(500 + 5 \sqrt{9970})}^{3}}{3 \cdot 2250} \cdot 3 - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.4.3

Rewrite the expression.

$\frac{\frac{{(500 + 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.5

Use the Binomial Theorem.

$\frac{\frac{500^{3} + 3 \cdot 500^{2} (5 \sqrt{9970}) + 3 \cdot 500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6

Simplify each term.

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Step 4.1.2.4.6.1

Raise $500$ to the power of $3$ .

$\frac{\frac{125000000 + 3 \cdot 500^{2} (5 \sqrt{9970}) + 3 \cdot 500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.2

Raise $500$ to the power of $2$ .

$\frac{\frac{125000000 + 3 \cdot 250000 (5 \sqrt{9970}) + 3 \cdot 500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.3

Multiply $3$ by $250000$ .

$\frac{\frac{125000000 + 750000 (5 \sqrt{9970}) + 3 \cdot 500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.4

Multiply $5$ by $750000$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 3 \cdot 500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.5

Multiply $3$ by $500$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 {(5 \sqrt{9970})}^{2} + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.6

Apply the product rule to $5 \sqrt{9970}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (5^{2} {\sqrt{9970}}^{2}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.7

Raise $5$ to the power of $2$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 {\sqrt{9970}}^{2}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.8

Rewrite ${\sqrt{9970}}^{2}$ as $9970$ .

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Step 4.1.2.4.6.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9970}$ as $9970^{\frac{1}{2}}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 {(9970^{\frac{1}{2}})}^{2}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{\frac{1}{2} \cdot 2}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.8.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{\frac{2}{2}}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.8.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.6.8.4.1

Cancel the common factor.

Step 4.1.2.4.6.8.4.2

Rewrite the expression.

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{1}) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.8.5

Evaluate the exponent.

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 (25 \cdot 9970) + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.9

Multiply $1500 (25 \cdot 9970)$ .

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Step 4.1.2.4.6.9.1

Multiply $25$ by $9970$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 1500 \cdot 249250 + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.9.2

Multiply $1500$ by $249250$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + {(5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.10

Apply the product rule to $5 \sqrt{9970}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 5^{3} {\sqrt{9970}}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.11

Raise $5$ to the power of $3$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 {\sqrt{9970}}^{3}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.12

Rewrite ${\sqrt{9970}}^{3}$ as $\sqrt{9970^{3}}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 \sqrt{9970^{3}}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.13

Raise $9970$ to the power of $3$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 \sqrt{991026973000}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.14

Rewrite $991026973000$ as $9970^{2} \cdot 9970$ .

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Step 4.1.2.4.6.14.1

Factor $99400900$ out of $991026973000$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 \sqrt{99400900 (9970)}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.14.2

Rewrite $99400900$ as $9970^{2}$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 \sqrt{9970^{2} \cdot 9970}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.15

Pull terms out from under the radical.

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 125 (9970 \sqrt{9970})}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.6.16

Multiply $9970$ by $125$ .

$\frac{\frac{125000000 + 3750000 \sqrt{9970} + 373875000 + 1246250 \sqrt{9970}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.7

Add $125000000$ and $373875000$ .

$\frac{\frac{498875000 + 3750000 \sqrt{9970} + 1246250 \sqrt{9970}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.8

Add $3750000 \sqrt{9970}$ and $1246250 \sqrt{9970}$ .

$\frac{\frac{498875000 + 4996250 \sqrt{9970}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9

Cancel the common factor of $498875000 + 4996250 \sqrt{9970}$ and $2250$ .

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Step 4.1.2.4.9.1

Factor $250$ out of $498875000$ .

$\frac{\frac{250 (1995500) + 4996250 \sqrt{9970}}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9.2

Factor $250$ out of $4996250 \sqrt{9970}$ .

$\frac{\frac{250 (1995500) + 250 (19985 \sqrt{9970})}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9.3

Factor $250$ out of $250 (1995500) + 250 (19985 \sqrt{9970})$ .

$\frac{\frac{250 (1995500 + 19985 \sqrt{9970})}{2250} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9.4

Cancel the common factors.

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Step 4.1.2.4.9.4.1

Factor $250$ out of $2250$ .

$\frac{\frac{250 (1995500 + 19985 \sqrt{9970})}{250 \cdot 9} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9.4.2

Cancel the common factor.

$\frac{\frac{250 (1995500 + 19985 \sqrt{9970})}{250 \cdot 9} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.9.4.3

Rewrite the expression.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 {(\frac{500 + 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.10

Apply the product rule to $\frac{500 + 5 \sqrt{9970}}{3}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{{(500 + 5 \sqrt{9970})}^{2}}{3^{2}} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.11

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

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{{(500 + 5 \sqrt{9970})}^{2}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.12

Rewrite ${(500 + 5 \sqrt{9970})}^{2}$ as $(500 + 5 \sqrt{9970}) (500 + 5 \sqrt{9970})$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{(500 + 5 \sqrt{9970}) (500 + 5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.13

Expand $(500 + 5 \sqrt{9970}) (500 + 5 \sqrt{9970})$ using the FOIL Method.

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Step 4.1.2.4.13.1

Apply the distributive property.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{500 (500 + 5 \sqrt{9970}) + 5 \sqrt{9970} (500 + 5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.13.2

Apply the distributive property.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{500 \cdot 500 + 500 (5 \sqrt{9970}) + 5 \sqrt{9970} (500 + 5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.13.3

Apply the distributive property.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{500 \cdot 500 + 500 (5 \sqrt{9970}) + 5 \sqrt{9970} \cdot 500 + 5 \sqrt{9970} (5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14

Simplify and combine like terms.

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Step 4.1.2.4.14.1

Simplify each term.

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Step 4.1.2.4.14.1.1

Multiply $500$ by $500$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 500 (5 \sqrt{9970}) + 5 \sqrt{9970} \cdot 500 + 5 \sqrt{9970} (5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.2

Multiply $5$ by $500$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 5 \sqrt{9970} \cdot 500 + 5 \sqrt{9970} (5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.3

Multiply $500$ by $5$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 5 \sqrt{9970} (5 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.4

Multiply $5 \sqrt{9970} (5 \sqrt{9970})$ .

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Step 4.1.2.4.14.1.4.1

Multiply $5$ by $5$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \sqrt{9970} \sqrt{9970}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.4.2

Raise $\sqrt{9970}$ to the power of $1$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 ({\sqrt{9970}}^{1} \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.4.3

Raise $\sqrt{9970}$ to the power of $1$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 ({\sqrt{9970}}^{1} {\sqrt{9970}}^{1})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.4.4

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

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 {\sqrt{9970}}^{1 + 1}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.4.5

Add $1$ and $1$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 {\sqrt{9970}}^{2}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5

Rewrite ${\sqrt{9970}}^{2}$ as $9970$ .

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Step 4.1.2.4.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9970}$ as $9970^{\frac{1}{2}}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 {(9970^{\frac{1}{2}})}^{2}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{1}{2} \cdot 2}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{2}{2}}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.4.14.1.5.4.1

Cancel the common factor.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{2}{2}}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5.4.2

Rewrite the expression.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \cdot 9970^{1}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.5.5

Evaluate the exponent.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 25 \cdot 9970}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.1.6

Multiply $25$ by $9970$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 2500 \sqrt{9970} + 2500 \sqrt{9970} + 249250}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.2

Add $250000$ and $249250$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{499250 + 2500 \sqrt{9970} + 2500 \sqrt{9970}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.14.3

Add $2500 \sqrt{9970}$ and $2500 \sqrt{9970}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - 2 \frac{499250 + 5000 \sqrt{9970}}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.15

Combine $- 2$ and $\frac{499250 + 5000 \sqrt{9970}}{9}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 + 5000 \sqrt{9970})}{9} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.16

Cancel the common factor of $3$ .

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Step 4.1.2.4.16.1

Factor $3$ out of $9$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 + 5000 \sqrt{9970})}{3 (3)} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.16.2

Cancel the common factor.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 + 5000 \sqrt{9970})}{3 \cdot 3} \cdot 3 + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.16.3

Rewrite the expression.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 + 5000 \sqrt{9970})}{3} + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.17

Move the negative in front of the fraction.

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - \frac{2 (499250 + 5000 \sqrt{9970})}{3} + 500 + 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.1.2.4.18

Multiply $3$ by $3$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - \frac{2 (499250 + 5000 \sqrt{9970})}{3} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.5

To write $- \frac{2 (499250 + 5000 \sqrt{9970})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - \frac{2 (499250 + 5000 \sqrt{9970})}{3} \cdot \frac{3}{3} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.6

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.6.1

Multiply $\frac{2 (499250 + 5000 \sqrt{9970})}{3}$ by $\frac{3}{3}$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - \frac{2 (499250 + 5000 \sqrt{9970}) \cdot 3}{3 \cdot 3} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.6.2

Multiply $3$ by $3$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970}}{9} - \frac{2 (499250 + 5000 \sqrt{9970}) \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.7

Combine the numerators over the common denominator.

$\frac{\frac{1995500 + 19985 \sqrt{9970} - 2 (499250 + 5000 \sqrt{9970}) \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8

Simplify the numerator.

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Step 4.1.2.8.1

Apply the distributive property.

$\frac{\frac{1995500 + 19985 \sqrt{9970} + (- 2 \cdot 499250 - 2 (5000 \sqrt{9970})) \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.2

Multiply $- 2$ by $499250$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970} + (- 998500 - 2 (5000 \sqrt{9970})) \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.3

Multiply $5000$ by $- 2$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970} + (- 998500 - 10000 \sqrt{9970}) \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.4

Apply the distributive property.

$\frac{\frac{1995500 + 19985 \sqrt{9970} - 998500 \cdot 3 - 10000 \sqrt{9970} \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.5

Multiply $- 998500$ by $3$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970} - 2995500 - 10000 \sqrt{9970} \cdot 3}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.6

Multiply $3$ by $- 10000$ .

$\frac{\frac{1995500 + 19985 \sqrt{9970} - 2995500 - 30000 \sqrt{9970}}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.7

Subtract $2995500$ from $1995500$ .

$\frac{\frac{- 1000000 + 19985 \sqrt{9970} - 30000 \sqrt{9970}}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.8.8

Subtract $30000 \sqrt{9970}$ from $19985 \sqrt{9970}$ .

$\frac{\frac{- 1000000 - 10015 \sqrt{9970}}{9} + 500 + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.9

To write $500$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 1000000 - 10015 \sqrt{9970}}{9} + 500 \cdot \frac{9}{9} + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.10

Combine $500$ and $\frac{9}{9}$ .

$\frac{\frac{- 1000000 - 10015 \sqrt{9970}}{9} + \frac{500 \cdot 9}{9} + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.11

Simplify the expression.

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Step 4.1.2.11.1

Combine the numerators over the common denominator.

$\frac{\frac{- 1000000 - 10015 \sqrt{9970} + 500 \cdot 9}{9} + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.11.2

Multiply $500$ by $9$ .

$\frac{\frac{- 1000000 - 10015 \sqrt{9970} + 4500}{9} + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.11.3

Add $- 1000000$ and $4500$ .

$\frac{\frac{- 995500 - 10015 \sqrt{9970}}{9} + 5 \sqrt{9970} + 9}{3}$

Step 4.1.2.12

To write $5 \sqrt{9970}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 995500 - 10015 \sqrt{9970}}{9} + 5 \sqrt{9970} \cdot \frac{9}{9} + 9}{3}$

Step 4.1.2.13

Combine fractions.

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Step 4.1.2.13.1

Combine $5 \sqrt{9970}$ and $\frac{9}{9}$ .

$\frac{\frac{- 995500 - 10015 \sqrt{9970}}{9} + \frac{5 \sqrt{9970} \cdot 9}{9} + 9}{3}$

Step 4.1.2.13.2

Combine the numerators over the common denominator.

$\frac{\frac{- 995500 - 10015 \sqrt{9970} + 5 \sqrt{9970} \cdot 9}{9} + 9}{3}$

Step 4.1.2.14

Simplify the numerator.

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Step 4.1.2.14.1

Multiply $9$ by $5$ .

$\frac{\frac{- 995500 - 10015 \sqrt{9970} + 45 \sqrt{9970}}{9} + 9}{3}$

Step 4.1.2.14.2

Add $- 10015 \sqrt{9970}$ and $45 \sqrt{9970}$ .

$\frac{\frac{- 995500 - 9970 \sqrt{9970}}{9} + 9}{3}$

Step 4.1.2.15

To write $9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 995500 - 9970 \sqrt{9970}}{9} + 9 \cdot \frac{9}{9}}{3}$

Step 4.1.2.16

Combine fractions.

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Step 4.1.2.16.1

Combine $9$ and $\frac{9}{9}$ .

$\frac{\frac{- 995500 - 9970 \sqrt{9970}}{9} + \frac{9 \cdot 9}{9}}{3}$

Step 4.1.2.16.2

Combine the numerators over the common denominator.

$\frac{\frac{- 995500 - 9970 \sqrt{9970} + 9 \cdot 9}{9}}{3}$

Step 4.1.2.17

Simplify the numerator.

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Step 4.1.2.17.1

Multiply $9$ by $9$ .

$\frac{\frac{- 995500 - 9970 \sqrt{9970} + 81}{9}}{3}$

Step 4.1.2.17.2

Add $- 995500$ and $81$ .

$\frac{\frac{- 995419 - 9970 \sqrt{9970}}{9}}{3}$

Step 4.1.2.18

Simplify with factoring out.

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Step 4.1.2.18.1

Rewrite $- 995419$ as $- 1 (995419)$ .

$\frac{\frac{- 1 (995419) - 9970 \sqrt{9970}}{9}}{3}$

Step 4.1.2.18.2

Factor $- 1$ out of $- 9970 \sqrt{9970}$ .

$\frac{\frac{- 1 (995419) - (9970 \sqrt{9970})}{9}}{3}$

Step 4.1.2.18.3

Factor $- 1$ out of $- 1 (995419) - (9970 \sqrt{9970})$ .

$\frac{\frac{- 1 (995419 + 9970 \sqrt{9970})}{9}}{3}$

Step 4.1.2.18.4

Move the negative in front of the fraction.

$\frac{- \frac{995419 + 9970 \sqrt{9970}}{9}}{3}$

Step 4.1.2.19

Multiply the numerator by the reciprocal of the denominator.

$- \frac{995419 + 9970 \sqrt{9970}}{9} \cdot \frac{1}{3}$

Step 4.1.2.20

Multiply $- \frac{995419 + 9970 \sqrt{9970}}{9} \cdot \frac{1}{3}$ .

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Step 4.1.2.20.1

Multiply $\frac{1}{3}$ by $\frac{995419 + 9970 \sqrt{9970}}{9}$ .

$- \frac{995419 + 9970 \sqrt{9970}}{3 \cdot 9}$

Step 4.1.2.20.2

Multiply $3$ by $9$ .

$- \frac{995419 + 9970 \sqrt{9970}}{27}$

Step 4.2

Evaluate at $x = \frac{500 - 5 \sqrt{9970}}{3}$ .

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Step 4.2.1

Substitute $\frac{500 - 5 \sqrt{9970}}{3}$ for $x$ .

$0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Remove parentheses.

$0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2

Find the common denominator.

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Step 4.2.2.2.1

Write $0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3}$ as a fraction with denominator $1$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3}}{1} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.2

Multiply $\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3}}{1} \cdot \frac{3}{3} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.3

Multiply $\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.4

Write $- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2}$ as a fraction with denominator $1$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2}}{1} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.5

Multiply $\frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2}}{1} \cdot \frac{3}{3} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.6

Multiply $\frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 - 5 \sqrt{9970}}{3} + 3$

Step 4.2.2.2.7

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

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 - 5 \sqrt{9970}}{3} + \frac{3}{1}$

Step 4.2.2.2.8

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

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 - 5 \sqrt{9970}}{3} + \frac{3}{1} \cdot \frac{3}{3}$

Step 4.2.2.2.9

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

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3}{3} + \frac{- 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3}{3} + \frac{500 - 5 \sqrt{9970}}{3} + \frac{3 \cdot 3}{3}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$\frac{0.004 {(\frac{500 - 5 \sqrt{9970}}{3})}^{3} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4

Simplify each term.

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Step 4.2.2.4.1

Apply the product rule to $\frac{500 - 5 \sqrt{9970}}{3}$ .

$\frac{0.004 \frac{{(500 - 5 \sqrt{9970})}^{3}}{3^{3}} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.2

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

$\frac{0.004 \frac{{(500 - 5 \sqrt{9970})}^{3}}{27} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.3

Cancel the common factor of $0.004$ .

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Step 4.2.2.4.3.1

Factor $0.004$ out of $27$ .

$\frac{0.004 \frac{{(500 - 5 \sqrt{9970})}^{3}}{0.004 (6750)} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.3.2

Cancel the common factor.

$\frac{0.004 \frac{{(500 - 5 \sqrt{9970})}^{3}}{0.004 \cdot 6750} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.3.3

Rewrite the expression.

$\frac{\frac{{(500 - 5 \sqrt{9970})}^{3}}{6750} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.4

Cancel the common factor of $3$ .

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Step 4.2.2.4.4.1

Factor $3$ out of $6750$ .

$\frac{\frac{{(500 - 5 \sqrt{9970})}^{3}}{3 (2250)} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.4.2

Cancel the common factor.

$\frac{\frac{{(500 - 5 \sqrt{9970})}^{3}}{3 \cdot 2250} \cdot 3 - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.4.3

Rewrite the expression.

$\frac{\frac{{(500 - 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.5

Use the Binomial Theorem.

$\frac{\frac{500^{3} + 3 \cdot 500^{2} (- 5 \sqrt{9970}) + 3 \cdot 500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6

Simplify each term.

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Step 4.2.2.4.6.1

Raise $500$ to the power of $3$ .

$\frac{\frac{125000000 + 3 \cdot 500^{2} (- 5 \sqrt{9970}) + 3 \cdot 500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.2

Raise $500$ to the power of $2$ .

$\frac{\frac{125000000 + 3 \cdot 250000 (- 5 \sqrt{9970}) + 3 \cdot 500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.3

Multiply $3$ by $250000$ .

$\frac{\frac{125000000 + 750000 (- 5 \sqrt{9970}) + 3 \cdot 500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.4

Multiply $- 5$ by $750000$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 3 \cdot 500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.5

Multiply $3$ by $500$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 {(- 5 \sqrt{9970})}^{2} + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.6

Apply the product rule to $- 5 \sqrt{9970}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 ({(- 5)}^{2} {\sqrt{9970}}^{2}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.7

Raise $- 5$ to the power of $2$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 {\sqrt{9970}}^{2}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.8

Rewrite ${\sqrt{9970}}^{2}$ as $9970$ .

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Step 4.2.2.4.6.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9970}$ as $9970^{\frac{1}{2}}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 {(9970^{\frac{1}{2}})}^{2}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{\frac{1}{2} \cdot 2}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.8.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{\frac{2}{2}}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.8.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.6.8.4.1

Cancel the common factor.

Step 4.2.2.4.6.8.4.2

Rewrite the expression.

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 \cdot 9970^{1}) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.8.5

Evaluate the exponent.

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 (25 \cdot 9970) + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.9

Multiply $1500 (25 \cdot 9970)$ .

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Step 4.2.2.4.6.9.1

Multiply $25$ by $9970$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 1500 \cdot 249250 + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.9.2

Multiply $1500$ by $249250$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 + {(- 5 \sqrt{9970})}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.10

Apply the product rule to $- 5 \sqrt{9970}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 + {(- 5)}^{3} {\sqrt{9970}}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.11

Raise $- 5$ to the power of $3$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 {\sqrt{9970}}^{3}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.12

Rewrite ${\sqrt{9970}}^{3}$ as $\sqrt{9970^{3}}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 \sqrt{9970^{3}}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.13

Raise $9970$ to the power of $3$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 \sqrt{991026973000}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.14

Rewrite $991026973000$ as $9970^{2} \cdot 9970$ .

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Step 4.2.2.4.6.14.1

Factor $99400900$ out of $991026973000$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 \sqrt{99400900 (9970)}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.14.2

Rewrite $99400900$ as $9970^{2}$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 \sqrt{9970^{2} \cdot 9970}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.15

Pull terms out from under the radical.

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 125 (9970 \sqrt{9970})}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.6.16

Multiply $9970$ by $- 125$ .

$\frac{\frac{125000000 - 3750000 \sqrt{9970} + 373875000 - 1246250 \sqrt{9970}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.7

Add $125000000$ and $373875000$ .

$\frac{\frac{498875000 - 3750000 \sqrt{9970} - 1246250 \sqrt{9970}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.8

Subtract $1246250 \sqrt{9970}$ from $- 3750000 \sqrt{9970}$ .

$\frac{\frac{498875000 - 4996250 \sqrt{9970}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9

Cancel the common factor of $498875000 - 4996250 \sqrt{9970}$ and $2250$ .

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Step 4.2.2.4.9.1

Factor $250$ out of $498875000$ .

$\frac{\frac{250 (1995500) - 4996250 \sqrt{9970}}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9.2

Factor $250$ out of $- 4996250 \sqrt{9970}$ .

$\frac{\frac{250 (1995500) + 250 (- 19985 \sqrt{9970})}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9.3

Factor $250$ out of $250 (1995500) + 250 (- 19985 \sqrt{9970})$ .

$\frac{\frac{250 (1995500 - 19985 \sqrt{9970})}{2250} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9.4

Cancel the common factors.

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Step 4.2.2.4.9.4.1

Factor $250$ out of $2250$ .

$\frac{\frac{250 (1995500 - 19985 \sqrt{9970})}{250 \cdot 9} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9.4.2

Cancel the common factor.

$\frac{\frac{250 (1995500 - 19985 \sqrt{9970})}{250 \cdot 9} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.9.4.3

Rewrite the expression.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 {(\frac{500 - 5 \sqrt{9970}}{3})}^{2} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.10

Apply the product rule to $\frac{500 - 5 \sqrt{9970}}{3}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{{(500 - 5 \sqrt{9970})}^{2}}{3^{2}} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.11

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

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{{(500 - 5 \sqrt{9970})}^{2}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.12

Rewrite ${(500 - 5 \sqrt{9970})}^{2}$ as $(500 - 5 \sqrt{9970}) (500 - 5 \sqrt{9970})$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{(500 - 5 \sqrt{9970}) (500 - 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.13

Expand $(500 - 5 \sqrt{9970}) (500 - 5 \sqrt{9970})$ using the FOIL Method.

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Step 4.2.2.4.13.1

Apply the distributive property.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{500 (500 - 5 \sqrt{9970}) - 5 \sqrt{9970} (500 - 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.13.2

Apply the distributive property.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{500 \cdot 500 + 500 (- 5 \sqrt{9970}) - 5 \sqrt{9970} (500 - 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.13.3

Apply the distributive property.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{500 \cdot 500 + 500 (- 5 \sqrt{9970}) - 5 \sqrt{9970} \cdot 500 - 5 \sqrt{9970} (- 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14

Simplify and combine like terms.

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Step 4.2.2.4.14.1

Simplify each term.

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Step 4.2.2.4.14.1.1

Multiply $500$ by $500$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 + 500 (- 5 \sqrt{9970}) - 5 \sqrt{9970} \cdot 500 - 5 \sqrt{9970} (- 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.2

Multiply $- 5$ by $500$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 5 \sqrt{9970} \cdot 500 - 5 \sqrt{9970} (- 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.3

Multiply $500$ by $- 5$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} - 5 \sqrt{9970} (- 5 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.4

Multiply $- 5 \sqrt{9970} (- 5 \sqrt{9970})$ .

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Step 4.2.2.4.14.1.4.1

Multiply $- 5$ by $- 5$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \sqrt{9970} \sqrt{9970}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.4.2

Raise $\sqrt{9970}$ to the power of $1$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 ({\sqrt{9970}}^{1} \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.4.3

Raise $\sqrt{9970}$ to the power of $1$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 ({\sqrt{9970}}^{1} {\sqrt{9970}}^{1})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.4.4

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

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 {\sqrt{9970}}^{1 + 1}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.4.5

Add $1$ and $1$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 {\sqrt{9970}}^{2}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5

Rewrite ${\sqrt{9970}}^{2}$ as $9970$ .

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Step 4.2.2.4.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9970}$ as $9970^{\frac{1}{2}}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 {(9970^{\frac{1}{2}})}^{2}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{1}{2} \cdot 2}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{2}{2}}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.4.14.1.5.4.1

Cancel the common factor.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \cdot 9970^{\frac{2}{2}}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5.4.2

Rewrite the expression.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \cdot 9970^{1}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.5.5

Evaluate the exponent.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 25 \cdot 9970}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.1.6

Multiply $25$ by $9970$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{250000 - 2500 \sqrt{9970} - 2500 \sqrt{9970} + 249250}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.2

Add $250000$ and $249250$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{499250 - 2500 \sqrt{9970} - 2500 \sqrt{9970}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.14.3

Subtract $2500 \sqrt{9970}$ from $- 2500 \sqrt{9970}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - 2 \frac{499250 - 5000 \sqrt{9970}}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.15

Combine $- 2$ and $\frac{499250 - 5000 \sqrt{9970}}{9}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 - 5000 \sqrt{9970})}{9} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.16

Cancel the common factor of $3$ .

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Step 4.2.2.4.16.1

Factor $3$ out of $9$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 - 5000 \sqrt{9970})}{3 (3)} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.16.2

Cancel the common factor.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 - 5000 \sqrt{9970})}{3 \cdot 3} \cdot 3 + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.16.3

Rewrite the expression.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} + \frac{- 2 (499250 - 5000 \sqrt{9970})}{3} + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.17

Move the negative in front of the fraction.

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - \frac{2 (499250 - 5000 \sqrt{9970})}{3} + 500 - 5 \sqrt{9970} + 3 \cdot 3}{3}$

Step 4.2.2.4.18

Multiply $3$ by $3$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - \frac{2 (499250 - 5000 \sqrt{9970})}{3} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.5

To write $- \frac{2 (499250 - 5000 \sqrt{9970})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - \frac{2 (499250 - 5000 \sqrt{9970})}{3} \cdot \frac{3}{3} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.6

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.6.1

Multiply $\frac{2 (499250 - 5000 \sqrt{9970})}{3}$ by $\frac{3}{3}$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - \frac{2 (499250 - 5000 \sqrt{9970}) \cdot 3}{3 \cdot 3} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.6.2

Multiply $3$ by $3$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970}}{9} - \frac{2 (499250 - 5000 \sqrt{9970}) \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.7

Combine the numerators over the common denominator.

$\frac{\frac{1995500 - 19985 \sqrt{9970} - 2 (499250 - 5000 \sqrt{9970}) \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8

Simplify the numerator.

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Step 4.2.2.8.1

Apply the distributive property.

$\frac{\frac{1995500 - 19985 \sqrt{9970} + (- 2 \cdot 499250 - 2 (- 5000 \sqrt{9970})) \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.2

Multiply $- 2$ by $499250$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970} + (- 998500 - 2 (- 5000 \sqrt{9970})) \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.3

Multiply $- 5000$ by $- 2$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970} + (- 998500 + 10000 \sqrt{9970}) \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.4

Apply the distributive property.

$\frac{\frac{1995500 - 19985 \sqrt{9970} - 998500 \cdot 3 + 10000 \sqrt{9970} \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.5

Multiply $- 998500$ by $3$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970} - 2995500 + 10000 \sqrt{9970} \cdot 3}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.6

Multiply $3$ by $10000$ .

$\frac{\frac{1995500 - 19985 \sqrt{9970} - 2995500 + 30000 \sqrt{9970}}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.7

Subtract $2995500$ from $1995500$ .

$\frac{\frac{- 1000000 - 19985 \sqrt{9970} + 30000 \sqrt{9970}}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.8.8

Add $- 19985 \sqrt{9970}$ and $30000 \sqrt{9970}$ .

$\frac{\frac{- 1000000 + 10015 \sqrt{9970}}{9} + 500 - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.9

To write $500$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 1000000 + 10015 \sqrt{9970}}{9} + 500 \cdot \frac{9}{9} - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.10

Combine $500$ and $\frac{9}{9}$ .

$\frac{\frac{- 1000000 + 10015 \sqrt{9970}}{9} + \frac{500 \cdot 9}{9} - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.11

Simplify the expression.

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Step 4.2.2.11.1

Combine the numerators over the common denominator.

$\frac{\frac{- 1000000 + 10015 \sqrt{9970} + 500 \cdot 9}{9} - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.11.2

Multiply $500$ by $9$ .

$\frac{\frac{- 1000000 + 10015 \sqrt{9970} + 4500}{9} - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.11.3

Add $- 1000000$ and $4500$ .

$\frac{\frac{- 995500 + 10015 \sqrt{9970}}{9} - 5 \sqrt{9970} + 9}{3}$

Step 4.2.2.12

To write $- 5 \sqrt{9970}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 995500 + 10015 \sqrt{9970}}{9} - 5 \sqrt{9970} \cdot \frac{9}{9} + 9}{3}$

Step 4.2.2.13

Combine fractions.

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Step 4.2.2.13.1

Combine $- 5 \sqrt{9970}$ and $\frac{9}{9}$ .

$\frac{\frac{- 995500 + 10015 \sqrt{9970}}{9} + \frac{- 5 \sqrt{9970} \cdot 9}{9} + 9}{3}$

Step 4.2.2.13.2

Combine the numerators over the common denominator.

$\frac{\frac{- 995500 + 10015 \sqrt{9970} - 5 \sqrt{9970} \cdot 9}{9} + 9}{3}$

Step 4.2.2.14

Simplify the numerator.

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Step 4.2.2.14.1

Multiply $9$ by $- 5$ .

$\frac{\frac{- 995500 + 10015 \sqrt{9970} - 45 \sqrt{9970}}{9} + 9}{3}$

Step 4.2.2.14.2

Subtract $45 \sqrt{9970}$ from $10015 \sqrt{9970}$ .

$\frac{\frac{- 995500 + 9970 \sqrt{9970}}{9} + 9}{3}$

Step 4.2.2.15

To write $9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 995500 + 9970 \sqrt{9970}}{9} + 9 \cdot \frac{9}{9}}{3}$

Step 4.2.2.16

Combine fractions.

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Step 4.2.2.16.1

Combine $9$ and $\frac{9}{9}$ .

$\frac{\frac{- 995500 + 9970 \sqrt{9970}}{9} + \frac{9 \cdot 9}{9}}{3}$

Step 4.2.2.16.2

Combine the numerators over the common denominator.

$\frac{\frac{- 995500 + 9970 \sqrt{9970} + 9 \cdot 9}{9}}{3}$

Step 4.2.2.17

Simplify the numerator.

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Step 4.2.2.17.1

Multiply $9$ by $9$ .

$\frac{\frac{- 995500 + 9970 \sqrt{9970} + 81}{9}}{3}$

Step 4.2.2.17.2

Add $- 995500$ and $81$ .

$\frac{\frac{- 995419 + 9970 \sqrt{9970}}{9}}{3}$

Step 4.2.2.18

Simplify with factoring out.

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Step 4.2.2.18.1

Rewrite $- 995419$ as $- 1 (995419)$ .

$\frac{\frac{- 1 (995419) + 9970 \sqrt{9970}}{9}}{3}$

Step 4.2.2.18.2

Factor $- 1$ out of $9970 \sqrt{9970}$ .

$\frac{\frac{- 1 (995419) - (- 9970 \sqrt{9970})}{9}}{3}$

Step 4.2.2.18.3

Factor $- 1$ out of $- 1 (995419) - (- 9970 \sqrt{9970})$ .

$\frac{\frac{- 1 (995419 - 9970 \sqrt{9970})}{9}}{3}$

Step 4.2.2.18.4

Move the negative in front of the fraction.

$\frac{- \frac{995419 - 9970 \sqrt{9970}}{9}}{3}$

Step 4.2.2.19

Multiply the numerator by the reciprocal of the denominator.

$- \frac{995419 - 9970 \sqrt{9970}}{9} \cdot \frac{1}{3}$

Step 4.2.2.20

Multiply $- \frac{995419 - 9970 \sqrt{9970}}{9} \cdot \frac{1}{3}$ .

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Step 4.2.2.20.1

Multiply $\frac{1}{3}$ by $\frac{995419 - 9970 \sqrt{9970}}{9}$ .

$- \frac{995419 - 9970 \sqrt{9970}}{3 \cdot 9}$

Step 4.2.2.20.2

Multiply $3$ by $9$ .

$- \frac{995419 - 9970 \sqrt{9970}}{27}$

Step 4.3

List all of the points.

$(\frac{500 + 5 \sqrt{9970}}{3}, - \frac{995419 + 9970 \sqrt{9970}}{27}), (\frac{500 - 5 \sqrt{9970}}{3}, - \frac{995419 - 9970 \sqrt{9970}}{27})$

Step 5