Basic Math Examples

$mg h = \frac{1}{2} \cdot (m v^{2})$

Step 1

Rewrite the equation as

$\frac{1}{2} \cdot (m v^{2}) = mg h$ .

$\frac{1}{2} \cdot (m v^{2}) = mg h$

Step 2

Multiply both sides of the equation by

$2$ .

$2 (\frac{1}{2} \cdot (m v^{2})) = 2 (mg h)$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

$2 (\frac{1}{2} \cdot (m v^{2}))$ .

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Step 3.1.1.1

Multiply

$\frac{1}{2} (m v^{2})$ .

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Step 3.1.1.1.1

Combine

$m$ and

$\frac{1}{2}$ .

$2 (\frac{m}{2} v^{2}) = 2 (mg h)$

Step 3.1.1.1.2

Combine

$\frac{m}{2}$ and

$v^{2}$ .

$2 \frac{m v^{2}}{2} = 2 (mg h)$

Step 3.1.1.2

Cancel the common factor of

$2$ .

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Step 3.1.1.2.1

Cancel the common factor.

$2 \frac{m v^{2}}{2} = 2 (mg h)$

Step 3.1.1.2.2

Rewrite the expression.

$m v^{2} = 2 (mg h)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Remove parentheses.

$m v^{2} = 2 mg h$

Step 4

Divide each term in

$m v^{2} = 2 mg h$ by

$m$ and simplify.

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Step 4.1

Divide each term in

$m v^{2} = 2 mg h$ by

$m$ .

$\frac{m v^{2}}{m} = \frac{2 mg h}{m}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

$m$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{m v^{2}}{m} = \frac{2 mg h}{m}$

Step 4.2.1.2

Divide

$v^{2}$ by

$1$ .

$v^{2} = \frac{2 mg h}{m}$

Step 5

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

$v = \pm \sqrt{\frac{2 mg h}{m}}$

Step 6

Simplify

$\pm \sqrt{\frac{2 mg h}{m}}$ .

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Step 6.1

Rewrite

$\sqrt{\frac{2 mg h}{m}}$ as

$\frac{\sqrt{2 mg h}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 mg h}}{\sqrt{m}}$

Step 6.2

Multiply

$\frac{\sqrt{2 mg h}}{\sqrt{m}}$ by

$\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 mg h}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply

$\frac{\sqrt{2 mg h}}{\sqrt{m}}$ by

$\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

Step 6.3.2

Raise

$\sqrt{m}$ to the power of

$1$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

Step 6.3.3

Raise

$\sqrt{m}$ to the power of

$1$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

Step 6.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Step 6.3.5

Add

$1$ and

$1$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{{\sqrt{m}}^{2}}$

Step 6.3.6

Rewrite

${\sqrt{m}}^{2}$ as

$m$ .

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Step 6.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{m}$ as

$m^{\frac{1}{2}}$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

Step 6.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

Step 6.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 6.3.6.4

Cancel the common factor of

$2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 6.3.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{m^{1}}$

Step 6.3.6.5

Simplify.

$v = \pm \frac{\sqrt{2 mg h} \sqrt{m}}{m}$

Step 6.4

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{2 mg h m}}{m}$

Step 7

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

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Step 7.1

First, use the positive value of the

$\pm$ to find the first solution.

$v = \frac{\sqrt{2 mg h m}}{m}$

Step 7.2

Next, use the negative value of the

$\pm$ to find the second solution.

$v = - \frac{\sqrt{2 mg h m}}{m}$

Step 7.3

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

$v = \frac{\sqrt{2 mg h m}}{m}, - \frac{\sqrt{2 mg h m}}{m}$

Step 8

Reorder factors in

$v = \frac{\sqrt{2 mg h m}}{m}, - \frac{\sqrt{2 mg h m}}{m}$ .

$v = \frac{\sqrt{2 h m mg}}{m}, - \frac{\sqrt{2 h m mg}}{m}$