Pre-Algebra Examples

$\frac{1}{2} \cdot ((5 x) (x)) = 4000$

Step 1

Multiply both sides of the equation by $\frac{1}{\frac{1}{2} \cdot 5}$ .

$\frac{1}{\frac{1}{2} \cdot 5} (\frac{1}{2} \cdot (5 x \cdot x)) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{1}{\frac{1}{2} \cdot 5} (\frac{1}{2} \cdot (5 x \cdot x))$ .

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Step 2.1.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.1.1.1

Move $x$ .

$\frac{1}{\frac{1}{2} \cdot 5} (\frac{1}{2} \cdot (5 (x \cdot x))) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.1.2

Multiply $x$ by $x$ .

$\frac{1}{\frac{1}{2} \cdot 5} (\frac{1}{2} \cdot (5 x^{2})) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.2

Combine $\frac{1}{2}$ and $5$ .

$\frac{1}{\frac{5}{2}} (\frac{1}{2} \cdot (5 x^{2})) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{2}{5}) (\frac{1}{2} \cdot (5 x^{2})) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.4

Multiply $\frac{2}{5}$ by $1$ .

$\frac{2}{5} (\frac{1}{2} \cdot (5 x^{2})) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.5

Cancel the common factor of $5$ .

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Step 2.1.1.5.1

Factor $5$ out of $\frac{1}{2} \cdot (5 x^{2})$ .

$\frac{2}{5} (5 (\frac{1}{2} \cdot (x^{2}))) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.5.2

Cancel the common factor.

$\frac{2}{5} (5 (\frac{1}{2} \cdot (x^{2}))) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.5.3

Rewrite the expression.

$2 (\frac{1}{2} \cdot (x^{2})) = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.6

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

$2 \frac{x^{2}}{2} = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.7

Combine $2$ and $\frac{x^{2}}{2}$ .

$\frac{2 x^{2}}{2} = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.8

Cancel the common factor of $2$ .

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Step 2.1.1.8.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.1.1.8.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1}{\frac{1}{2} \cdot 5} \cdot 4000$ .

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Step 2.2.1.1

Combine $\frac{1}{2}$ and $5$ .

$x^{2} = \frac{1}{\frac{5}{2}} \cdot 4000$

Step 2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$x^{2} = 1 (\frac{2}{5}) \cdot 4000$

Step 2.2.1.3

Multiply $\frac{2}{5}$ by $1$ .

$x^{2} = \frac{2}{5} \cdot 4000$

Step 2.2.1.4

Cancel the common factor of $5$ .

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Step 2.2.1.4.1

Factor $5$ out of $4000$ .

$x^{2} = \frac{2}{5} \cdot (5 (800))$

Step 2.2.1.4.2

Cancel the common factor.

$x^{2} = \frac{2}{5} \cdot (5 \cdot 800)$

Step 2.2.1.4.3

Rewrite the expression.

$x^{2} = 2 \cdot 800$

Step 2.2.1.5

Multiply $2$ by $800$ .

$x^{2} = 1600$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1600}$

Step 4

Simplify $\pm \sqrt{1600}$ .

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Step 4.1

Rewrite $1600$ as $40^{2}$ .

$x = \pm \sqrt{40^{2}}$

Step 4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 40$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 40$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 40$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 40, - 40$