Solve for l 2.4=2pi square root of l/32

$2.4 = 2 π \sqrt{\frac{l}{32}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{l}{32}} = 2.4$ .

$2 π \sqrt{\frac{l}{32}} = 2.4$

Step 2

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

${(2 π \sqrt{\frac{l}{32}})}^{2} = {2.4}^{2}$

Step 3

Simplify each side of the equation.

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

${(2 π {(\frac{l}{32})}^{\frac{1}{2}})}^{2} = {2.4}^{2}$

Simplify the left side.

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

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

${(2 π \frac{l^{\frac{1}{2}}}{32^{\frac{1}{2}}})}^{2} = {2.4}^{2}$

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

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

${(\frac{l^{\frac{1}{2}} \cdot 2}{32^{\frac{1}{2}}} π)}^{2} = {2.4}^{2}$

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

${(\frac{l^{\frac{1}{2}} \cdot 2 π}{32^{\frac{1}{2}}})}^{2} = {2.4}^{2}$

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

${(\frac{2 \cdot l^{\frac{1}{2}} π}{32^{\frac{1}{2}}})}^{2} = {2.4}^{2}$

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

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

$\frac{{(2 l^{\frac{1}{2}} π)}^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

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

$\frac{{(2 l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

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

$\frac{2^{2} {(l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

Simplify the numerator.

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

$\frac{4 {(l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

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

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

$\frac{4 l^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 l^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

Rewrite the expression.

$\frac{4 l^{1} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

Simplify.

$\frac{4 l π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {2.4}^{2}$

Simplify the denominator.

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

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

$\frac{4 l π^{2}}{32^{\frac{1}{2} \cdot 2}} = {2.4}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 l π^{2}}{32^{\frac{1}{2} \cdot 2}} = {2.4}^{2}$

Rewrite the expression.

$\frac{4 l π^{2}}{32^{1}} = {2.4}^{2}$

Evaluate the exponent.

$\frac{4 l π^{2}}{32} = {2.4}^{2}$

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

Factor $4$ out of $4 l π^{2}$ .

$\frac{4 (l π^{2})}{32} = {2.4}^{2}$

Cancel the common factors.

Factor $4$ out of $32$ .

$\frac{4 (l π^{2})}{4 \cdot 8} = {2.4}^{2}$

Cancel the common factor.

$\frac{4 (l π^{2})}{4 \cdot 8} = {2.4}^{2}$

Rewrite the expression.

$\frac{l π^{2}}{8} = {2.4}^{2}$

Simplify the right side.

Raise $2.4$ to the power of $2$ .

$\frac{l π^{2}}{8} = 5.76$

Step 4

Solve for $l$ .

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

$\frac{8}{π^{2}} \cdot \frac{l π^{2}}{8} = \frac{8}{π^{2}} \cdot 5.76$

Simplify both sides of the equation.

Simplify the left side.

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

Combine.

$\frac{8 (l π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 5.76$

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 (l π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 5.76$

Rewrite the expression.

$\frac{l π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 5.76$

Cancel the common factor of $π^{2}$ .

Cancel the common factor.

$\frac{l π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 5.76$

Divide $l$ by $1$ .

$l = \frac{8}{π^{2}} \cdot 5.76$

Simplify the right side.

Simplify $\frac{8}{π^{2}} \cdot 5.76$ .

Multiply $\frac{8}{π^{2}} \cdot 5.76$ .

Combine $\frac{8}{π^{2}}$ and $5.76$ .

$l = \frac{8 \cdot 5.76}{π^{2}}$

Multiply $8$ by $5.76$ .

$l = \frac{46.08}{π^{2}}$

Replace $π$ with an approximation.

$l = \frac{46.08}{{3.14159265}^{2}}$

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

$l = \frac{46.08}{9.8696044}$

Divide $46.08$ by $9.8696044$ .

$l = 4.66888014$

Solve for l 2.4=2pi square root of l/32

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