Solve for r p=4piIr^2

$p = 4 π I r^{2}$

Step 1

Rewrite the equation as $4 π I r^{2} = p$ .

$4 π I r^{2} = p$

Step 2

Divide each term in $4 π I r^{2} = p$ by $4 π I$ and simplify.

Divide each term in $4 π I r^{2} = p$ by $4 π I$ .

$\frac{4 π I r^{2}}{4 π I} = \frac{p}{4 π I}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 π I r^{2}}{4 π I} = \frac{p}{4 π I}$

Rewrite the expression.

$\frac{π I r^{2}}{π I} = \frac{p}{4 π I}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π I r^{2}}{π I} = \frac{p}{4 π I}$

Rewrite the expression.

$\frac{I r^{2}}{I} = \frac{p}{4 π I}$

Cancel the common factor of $I$ .

Cancel the common factor.

$\frac{I r^{2}}{I} = \frac{p}{4 π I}$

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

$r^{2} = \frac{p}{4 π I}$

Step 3

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

$r = \pm \sqrt{\frac{p}{4 π I}}$

Step 4

Simplify $\pm \sqrt{\frac{p}{4 π I}}$ .

Rewrite $\frac{p}{4 π I}$ as ${(\frac{1}{2})}^{2} \frac{p}{π I}$ .

Factor the perfect power $1^{2}$ out of $p$ .

$r = \pm \sqrt{\frac{1^{2} p}{4 π I}}$

Factor the perfect power $2^{2}$ out of $4 π I$ .

$r = \pm \sqrt{\frac{1^{2} p}{2^{2} (π I)}}$

Rearrange the fraction $\frac{1^{2} p}{2^{2} (π I)}$ .

$r = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{p}{π I}}$

Pull terms out from under the radical.

$r = \pm \frac{1}{2} \sqrt{\frac{p}{π I}}$

Rewrite $\sqrt{\frac{p}{π I}}$ as $\frac{\sqrt{p}}{\sqrt{π I}}$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p}}{\sqrt{π I}}$

Multiply $\frac{\sqrt{p}}{\sqrt{π I}}$ by $\frac{\sqrt{π I}}{\sqrt{π I}}$ .

$r = \pm \frac{1}{2} (\frac{\sqrt{p}}{\sqrt{π I}} \cdot \frac{\sqrt{π I}}{\sqrt{π I}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{p}}{\sqrt{π I}}$ by $\frac{\sqrt{π I}}{\sqrt{π I}}$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{\sqrt{π I} \sqrt{π I}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{\sqrt{π I}}^{1} \sqrt{π I}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{\sqrt{π I}}^{1} {\sqrt{π I}}^{1}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{\sqrt{π I}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{\sqrt{π I}}^{2}}$

Rewrite ${\sqrt{π I}}^{2}$ as $π I$ .

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{({(π I)}^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{(π I)}^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{(π I)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{(π I)}^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{{(π I)}^{1}}$

Simplify.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p} \sqrt{π I}}{π I}$

Combine using the product rule for radicals.

$r = \pm \frac{1}{2} \cdot \frac{\sqrt{p π I}}{π I}$

Multiply $\frac{1}{2}$ by $\frac{\sqrt{p π I}}{π I}$ .

$r = \pm \frac{\sqrt{p π I}}{2 π I}$

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.

$r = \frac{\sqrt{p π I}}{2 π I}$

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

$r = - \frac{\sqrt{p π I}}{2 π I}$

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

$r = \frac{\sqrt{p π I}}{2 π I}$

$r = - \frac{\sqrt{p π I}}{2 π I}$

$r = \frac{\sqrt{p π I}}{2 π I}$

$r = - \frac{\sqrt{p π I}}{2 π I}$

Solve for r p=4piIr^2

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