Write as a Function of r r = square root of A/(4pi)

$r = \sqrt{\frac{A}{4 π}}$

Step 1

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

$\sqrt{\frac{A}{4 π}} = r$

Step 2

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

${\sqrt{\frac{A}{4 π}}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(\frac{A}{4 π})}^{1} = r^{2}$

Simplify.

$\frac{A}{4 π} = r^{2}$

Step 4

Solve for $A$ .

Multiply both sides of the equation by $4 π$ .

$4 π \frac{A}{4 π} = 4 π r^{2}$

Simplify the left side.

Cancel the common factor of $4 π$ .

Cancel the common factor.

$4 π \frac{A}{4 π} = 4 π r^{2}$

Rewrite the expression.

$A = 4 π r^{2}$

Step 5

To rewrite as a function of $r$ , write the equation so that $A$ is by itself on one side of the equal sign and an expression involving only $r$ is on the other side.

$f (r) = 4 π r^{2}$

Write as a Function of r r = square root of A/(4pi)

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