Solve for L 9.42=2pi square root of L/32

$9.42 = 2 π \sqrt{\frac{L}{32}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{L}{32}} = 9.42$ .

$2 π \sqrt{\frac{L}{32}} = 9.42$

Step 2

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

${(2 π \sqrt{\frac{L}{32}})}^{2} = {9.42}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{L}{32}}$ as ${(\frac{L}{32})}^{\frac{1}{2}}$ .

${(2 π {(\frac{L}{32})}^{\frac{1}{2}})}^{2} = {9.42}^{2}$

Simplify the left side.

Simplify ${(2 π {(\frac{L}{32})}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $\frac{L}{32}$ .

${(2 π \frac{L^{\frac{1}{2}}}{32^{\frac{1}{2}}})}^{2} = {9.42}^{2}$

Multiply $2 π \frac{L^{\frac{1}{2}}}{32^{\frac{1}{2}}}$ .

Combine $\frac{L^{\frac{1}{2}}}{32^{\frac{1}{2}}}$ and $2$ .

${(\frac{L^{\frac{1}{2}} \cdot 2}{32^{\frac{1}{2}}} π)}^{2} = {9.42}^{2}$

Combine $\frac{L^{\frac{1}{2}} \cdot 2}{32^{\frac{1}{2}}}$ and $π$ .

${(\frac{L^{\frac{1}{2}} \cdot 2 π}{32^{\frac{1}{2}}})}^{2} = {9.42}^{2}$

Move $2$ to the left of $L^{\frac{1}{2}}$ .

${(\frac{2 \cdot L^{\frac{1}{2}} π}{32^{\frac{1}{2}}})}^{2} = {9.42}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{2 L^{\frac{1}{2}} π}{32^{\frac{1}{2}}}$ .

$\frac{{(2 L^{\frac{1}{2}} π)}^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Apply the product rule to $2 L^{\frac{1}{2}} π$ .

$\frac{{(2 L^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Apply the product rule to $2 L^{\frac{1}{2}}$ .

$\frac{2^{2} {(L^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Simplify the numerator.

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

$\frac{4 {(L^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

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

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

$\frac{4 L^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 L^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Rewrite the expression.

$\frac{4 L^{1} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Simplify.

$\frac{4 L π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {9.42}^{2}$

Simplify the denominator.

Multiply the exponents in ${(32^{\frac{1}{2}})}^{2}$ .

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

$\frac{4 L π^{2}}{32^{\frac{1}{2} \cdot 2}} = {9.42}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 L π^{2}}{32^{\frac{1}{2} \cdot 2}} = {9.42}^{2}$

Rewrite the expression.

$\frac{4 L π^{2}}{32^{1}} = {9.42}^{2}$

Evaluate the exponent.

$\frac{4 L π^{2}}{32} = {9.42}^{2}$

Cancel the common factor of $4$ and $32$ .

Factor $4$ out of $4 L π^{2}$ .

$\frac{4 (L π^{2})}{32} = {9.42}^{2}$

Cancel the common factors.

Factor $4$ out of $32$ .

$\frac{4 (L π^{2})}{4 \cdot 8} = {9.42}^{2}$

Cancel the common factor.

$\frac{4 (L π^{2})}{4 \cdot 8} = {9.42}^{2}$

Rewrite the expression.

$\frac{L π^{2}}{8} = {9.42}^{2}$

Simplify the right side.

Raise $9.42$ to the power of $2$ .

$\frac{L π^{2}}{8} = 88.7364$

Step 4

Solve for $L$ .

Multiply both sides of the equation by $\frac{8}{π^{2}}$ .

$\frac{8}{π^{2}} \cdot \frac{L π^{2}}{8} = \frac{8}{π^{2}} \cdot 88.7364$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{8}{π^{2}} \cdot \frac{L π^{2}}{8}$ .

Combine.

$\frac{8 (L π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 88.7364$

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (L π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 88.7364$

Rewrite the expression.

$\frac{L π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 88.7364$

Cancel the common factor of $π^{2}$ .

Cancel the common factor.

$\frac{L π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 88.7364$

Divide $L$ by $1$ .

$L = \frac{8}{π^{2}} \cdot 88.7364$

Simplify the right side.

Simplify $\frac{8}{π^{2}} \cdot 88.7364$ .

Multiply $\frac{8}{π^{2}} \cdot 88.7364$ .

Combine $\frac{8}{π^{2}}$ and $88.7364$ .

$L = \frac{8 \cdot 88.7364}{π^{2}}$

Multiply $8$ by $88.7364$ .

$L = \frac{709.8912}{π^{2}}$

Replace $π$ with an approximation.

$L = \frac{709.8912}{{3.14159265}^{2}}$

Raise $3.14159265$ to the power of $2$ .

$L = \frac{709.8912}{9.8696044}$

Divide $709.8912$ by $9.8696044$ .

$L = 71.92701664$

Solve for L 9.42=2pi square root of L/32

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