Solve for r s=4pr^2

$s = 4 p r^{2}$

Step 1

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

$4 p r^{2} = s$

Step 2

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

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

$\frac{4 p r^{2}}{4 p} = \frac{s}{4 p}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 p r^{2}}{4 p} = \frac{s}{4 p}$

Rewrite the expression.

$\frac{p r^{2}}{p} = \frac{s}{4 p}$

Cancel the common factor of $p$ .

Cancel the common factor.

$\frac{p r^{2}}{p} = \frac{s}{4 p}$

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

$r^{2} = \frac{s}{4 p}$

Step 3

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

$r = \pm \sqrt{\frac{s}{4 p}}$

Step 4

Simplify $\pm \sqrt{\frac{s}{4 p}}$ .

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

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

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

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

$r = \pm \sqrt{\frac{1^{2} s}{2^{2} p}}$

Rearrange the fraction $\frac{1^{2} s}{2^{2} p}$ .

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

Pull terms out from under the radical.

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

Rewrite $\sqrt{\frac{s}{p}}$ as $\frac{\sqrt{s}}{\sqrt{p}}$ .

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

Multiply $\frac{\sqrt{s}}{\sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{s}}{\sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Combine using the product rule for radicals.

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

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

$r = \pm \frac{\sqrt{s p}}{2 p}$

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.

$r = \frac{\sqrt{s p}}{2 p}$

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

$r = - \frac{\sqrt{s p}}{2 p}$

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

$r = \frac{\sqrt{s p}}{2 p}$

$r = - \frac{\sqrt{s p}}{2 p}$

$r = \frac{\sqrt{s p}}{2 p}$

$r = - \frac{\sqrt{s p}}{2 p}$

Solve for r s=4pr^2

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