Solve for L r=2 square root of 5L

$r = 2 \sqrt{5 L}$

Step 1

Rewrite the equation as $2 \sqrt{5 L} = r$ .

$2 \sqrt{5 L} = r$

Step 2

Divide each term in $2 \sqrt{5 L} = r$ by $2$ and simplify.

Divide each term in $2 \sqrt{5 L} = r$ by $2$ .

$\frac{2 \sqrt{5 L}}{2} = \frac{r}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{5 L}}{2} = \frac{r}{2}$

Divide $\sqrt{5 L}$ by $1$ .

$\sqrt{5 L} = \frac{r}{2}$

Step 3

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

${\sqrt{5 L}}^{2} = {(\frac{r}{2})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 L}$ as ${(5 L)}^{\frac{1}{2}}$ .

${({(5 L)}^{\frac{1}{2}})}^{2} = {(\frac{r}{2})}^{2}$

Simplify the left side.

Simplify ${({(5 L)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 L)}^{\frac{1}{2}})}^{2}$ .

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

${(5 L)}^{\frac{1}{2} \cdot 2} = {(\frac{r}{2})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 L)}^{\frac{1}{2} \cdot 2} = {(\frac{r}{2})}^{2}$

Rewrite the expression.

${(5 L)}^{1} = {(\frac{r}{2})}^{2}$

Simplify.

$5 L = {(\frac{r}{2})}^{2}$

Simplify the right side.

Simplify ${(\frac{r}{2})}^{2}$ .

Apply the product rule to $\frac{r}{2}$ .

$5 L = \frac{r^{2}}{2^{2}}$

Raise $2$ to the power of $2$ .

$5 L = \frac{r^{2}}{4}$

Step 5

Divide each term in $5 L = \frac{r^{2}}{4}$ by $5$ and simplify.

Divide each term in $5 L = \frac{r^{2}}{4}$ by $5$ .

$\frac{5 L}{5} = \frac{\frac{r^{2}}{4}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 L}{5} = \frac{\frac{r^{2}}{4}}{5}$

Divide $L$ by $1$ .

$L = \frac{\frac{r^{2}}{4}}{5}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$L = \frac{r^{2}}{4} \cdot \frac{1}{5}$

Combine.

$L = \frac{r^{2} \cdot 1}{4 \cdot 5}$

Multiply.

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

$L = \frac{r^{2}}{4 \cdot 5}$

Multiply $4$ by $5$ .

$L = \frac{r^{2}}{20}$

Solve for L r=2 square root of 5L

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