Solve the Inequality for c E=mc^2

$E = m c^{2}$

Step 1

Rewrite the equation as $m c^{2} = E$ .

$m c^{2} = E$

Step 2

Divide each term in $m c^{2} = E$ by $m$ and simplify.

Divide each term in $m c^{2} = E$ by $m$ .

$\frac{m c^{2}}{m} = \frac{E}{m}$

Simplify the left side.

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m c^{2}}{m} = \frac{E}{m}$

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

$c^{2} = \frac{E}{m}$

Step 3

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

$c = \pm \sqrt{\frac{E}{m}}$

Step 4

Simplify $\pm \sqrt{\frac{E}{m}}$ .

Rewrite $\sqrt{\frac{E}{m}}$ as $\frac{\sqrt{E}}{\sqrt{m}}$ .

$c = \pm \frac{\sqrt{E}}{\sqrt{m}}$

Multiply $\frac{\sqrt{E}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$c = \pm \frac{\sqrt{E}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{E}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$c = \pm \frac{\sqrt{E} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Add $1$ and $1$ .

$c = \pm \frac{\sqrt{E} \sqrt{m}}{{\sqrt{m}}^{2}}$

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

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

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

$c = \pm \frac{\sqrt{E} \sqrt{m}}{m^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$c = \pm \frac{\sqrt{E} \sqrt{m}}{m^{\frac{2}{2}}}$

Rewrite the expression.

$c = \pm \frac{\sqrt{E} \sqrt{m}}{m^{1}}$

Simplify.

$c = \pm \frac{\sqrt{E} \sqrt{m}}{m}$

Combine using the product rule for radicals.

$c = \pm \frac{\sqrt{E m}}{m}$

Step 5

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

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

$c = \frac{\sqrt{E m}}{m}$

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

$c = - \frac{\sqrt{E m}}{m}$

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

$c = \frac{\sqrt{E m}}{m}$

$c = - \frac{\sqrt{E m}}{m}$

$c = \frac{\sqrt{E m}}{m}$

$c = - \frac{\sqrt{E m}}{m}$

Solve the Inequality for c E=mc^2

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