Solve for A r = square root of A/(4p)

$r = \sqrt{\frac{A}{4 p}}$

Step 1

Rewrite the equation as $\sqrt{\frac{A}{4 p}} = r$ .

$\sqrt{\frac{A}{4 p}} = r$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\frac{A}{4 p}}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{A}{4 p}}$ as ${(\frac{A}{4 p})}^{\frac{1}{2}}$ .

${({(\frac{A}{4 p})}^{\frac{1}{2}})}^{2} = r^{2}$

Simplify the left side.

Simplify ${({(\frac{A}{4 p})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{A}{4 p})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{A}{4 p})}^{\frac{1}{2} \cdot 2} = r^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{A}{4 p})}^{\frac{1}{2} \cdot 2} = r^{2}$

Rewrite the expression.

${(\frac{A}{4 p})}^{1} = r^{2}$

Simplify.

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

Step 4

Solve for $A$ .

Multiply both sides by $4 p$ .

$\frac{A}{4 p} (4 p) = r^{2} (4 p)$

Simplify.

Simplify the left side.

Simplify $\frac{A}{4 p} (4 p)$ .

Rewrite using the commutative property of multiplication.

$4 \frac{A}{4 p} p = r^{2} (4 p)$

Cancel the common factor of $4$ .

Factor $4$ out of $4 p$ .

$4 \frac{A}{4 (p)} p = r^{2} (4 p)$

Cancel the common factor.

$4 \frac{A}{4 p} p = r^{2} (4 p)$

Rewrite the expression.

$\frac{A}{p} p = r^{2} (4 p)$

Cancel the common factor of $p$ .

Cancel the common factor.

$\frac{A}{p} p = r^{2} (4 p)$

Rewrite the expression.

$A = r^{2} (4 p)$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$A = 4 r^{2} p$

Solve for A r = square root of A/(4p)

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