Algebra Examples

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

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

Step 1

Rewrite the equation as

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

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

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

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

Step 2

Multiply both sides of the equation by

2

$2$ .

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

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

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify

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

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

Tap for more steps...

Step 3.1.1

Multiply

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

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

Tap for more steps...

Step 3.1.1.1

Combine

m

$m$ and

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 3.1.1.2

Combine

\frac{m}{2}

$\frac{m}{2}$ and

v^{2}

$v^{2}$ .

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

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

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

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

Step 3.1.2

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.1.2.1

Cancel the common factor.

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

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

Step 3.1.2.2

Rewrite the expression.

m v^{2} = 2 h

$m v^{2} = 2 h$

m v^{2} = 2 h

$m v^{2} = 2 h$

m v^{2} = 2 h

$m v^{2} = 2 h$

m v^{2} = 2 h

$m v^{2} = 2 h$

Step 4

Divide each term in

m v^{2} = 2 h

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

m

$m$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in

m v^{2} = 2 h

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

m

$m$ .

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

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

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of

m

$m$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

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

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

Step 4.2.1.2

Divide

v^{2}

$v^{2}$ by

1

$1$ .

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

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

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

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

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

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

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

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

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 h}{m}}

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

Step 6

Simplify

\pm \sqrt{\frac{2 h}{m}}

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

Tap for more steps...

Step 6.1

Rewrite

\sqrt{\frac{2 h}{m}}

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

\frac{\sqrt{2 h}}{\sqrt{m}}

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

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

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

Step 6.2

Multiply

\frac{\sqrt{2 h}}{\sqrt{m}}

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

\frac{\sqrt{m}}{\sqrt{m}}

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

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

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

Step 6.3

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.1

Multiply

\frac{\sqrt{2 h}}{\sqrt{m}}

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

\frac{\sqrt{m}}{\sqrt{m}}

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

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

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

Step 6.3.2

Raise

\sqrt{m}

$\sqrt{m}$ to the power of

1

$1$ .

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

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

Step 6.3.3

Raise

\sqrt{m}

$\sqrt{m}$ to the power of

1

$1$ .

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

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

Step 6.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 6.3.5

Add

1

$1$ and

1

$1$ .

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

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

Step 6.3.6

Rewrite

{\sqrt{m}}^{2}

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

m

$m$ .

Tap for more steps...

Step 6.3.6.1

Use

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

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

\sqrt{m}

$\sqrt{m}$ as

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

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

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

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

Step 6.3.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 6.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 6.3.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 6.3.6.4.1

Cancel the common factor.

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

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

Step 6.3.6.4.2

Rewrite the expression.

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

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

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

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

Step 6.3.6.5

Simplify.

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

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

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

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

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

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

Step 6.4

Combine using the product rule for radicals.

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

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

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

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

Step 7

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

Tap for more steps...

Step 7.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

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

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

Step 7.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

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

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

Step 7.3

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

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

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

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

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

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

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

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

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