Algebra Examples

$3.4 = 2 π \sqrt{\frac{l}{32}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{l}{32}} = 3.4$ .

$2 π \sqrt{\frac{l}{32}} = 3.4$

Step 2

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

${(2 π \sqrt{\frac{l}{32}})}^{2} = {3.4}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

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

${(2 π {(\frac{l}{32})}^{\frac{1}{2}})}^{2} = {3.4}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

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

Tap for more steps...

Step 3.2.1.1

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

${(2 π \frac{l^{\frac{1}{2}}}{32^{\frac{1}{2}}})}^{2} = {3.4}^{2}$

Step 3.2.1.2

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

Tap for more steps...

Step 3.2.1.2.1

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

${(\frac{l^{\frac{1}{2}} \cdot 2}{32^{\frac{1}{2}}} π)}^{2} = {3.4}^{2}$

Step 3.2.1.2.2

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

${(\frac{l^{\frac{1}{2}} \cdot 2 π}{32^{\frac{1}{2}}})}^{2} = {3.4}^{2}$

Step 3.2.1.3

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

${(\frac{2 \cdot l^{\frac{1}{2}} π}{32^{\frac{1}{2}}})}^{2} = {3.4}^{2}$

Step 3.2.1.4

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

Tap for more steps...

Step 3.2.1.4.1

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

$\frac{{(2 l^{\frac{1}{2}} π)}^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.4.2

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

$\frac{{(2 l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.4.3

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

$\frac{2^{2} {(l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.5

Simplify the numerator.

Tap for more steps...

Step 3.2.1.5.1

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

$\frac{4 {(l^{\frac{1}{2}})}^{2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.5.2

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

Tap for more steps...

Step 3.2.1.5.2.1

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

$\frac{4 l^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.5.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.5.2.2.1

Cancel the common factor.

$\frac{4 l^{\frac{1}{2} \cdot 2} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.5.2.2.2

Rewrite the expression.

$\frac{4 l^{1} π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.5.3

Simplify.

$\frac{4 l π^{2}}{{(32^{\frac{1}{2}})}^{2}} = {3.4}^{2}$

Step 3.2.1.6

Simplify the denominator.

Tap for more steps...

Step 3.2.1.6.1

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

Tap for more steps...

Step 3.2.1.6.1.1

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

$\frac{4 l π^{2}}{32^{\frac{1}{2} \cdot 2}} = {3.4}^{2}$

Step 3.2.1.6.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.6.1.2.1

Cancel the common factor.

$\frac{4 l π^{2}}{32^{\frac{1}{2} \cdot 2}} = {3.4}^{2}$

Step 3.2.1.6.1.2.2

Rewrite the expression.

$\frac{4 l π^{2}}{32^{1}} = {3.4}^{2}$

Step 3.2.1.6.2

Evaluate the exponent.

$\frac{4 l π^{2}}{32} = {3.4}^{2}$

Step 3.2.1.7

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

Tap for more steps...

Step 3.2.1.7.1

Factor $4$ out of $4 l π^{2}$ .

$\frac{4 (l π^{2})}{32} = {3.4}^{2}$

Step 3.2.1.7.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1.7.2.1

Factor $4$ out of $32$ .

$\frac{4 (l π^{2})}{4 \cdot 8} = {3.4}^{2}$

Step 3.2.1.7.2.2

Cancel the common factor.

$\frac{4 (l π^{2})}{4 \cdot 8} = {3.4}^{2}$

Step 3.2.1.7.2.3

Rewrite the expression.

$\frac{l π^{2}}{8} = {3.4}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Raise $3.4$ to the power of $2$ .

$\frac{l π^{2}}{8} = 11.56$

Step 4

Solve for $l$ .

Tap for more steps...

Step 4.1

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

$\frac{8}{π^{2}} \cdot \frac{l π^{2}}{8} = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2

Simplify both sides of the equation.

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

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

Tap for more steps...

Step 4.2.1.1.1

Combine.

$\frac{8 (l π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2.1.1.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 4.2.1.1.2.1

Cancel the common factor.

$\frac{8 (l π^{2})}{π^{2} \cdot 8} = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2.1.1.2.2

Rewrite the expression.

$\frac{l π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2.1.1.3

Cancel the common factor of $π^{2}$ .

Tap for more steps...

Step 4.2.1.1.3.1

Cancel the common factor.

$\frac{l π^{2}}{π^{2}} = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2.1.1.3.2

Divide $l$ by $1$ .

$l = \frac{8}{π^{2}} \cdot 11.56$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Simplify $\frac{8}{π^{2}} \cdot 11.56$ .

Tap for more steps...

Step 4.2.2.1.1

Multiply $\frac{8}{π^{2}} \cdot 11.56$ .

Tap for more steps...

Step 4.2.2.1.1.1

Combine $\frac{8}{π^{2}}$ and $11.56$ .

$l = \frac{8 \cdot 11.56}{π^{2}}$

Step 4.2.2.1.1.2

Multiply $8$ by $11.56$ .

$l = \frac{92.48}{π^{2}}$

Step 4.2.2.1.2

Replace $π$ with an approximation.

$l = \frac{92.48}{{3.14159265}^{2}}$

Step 4.2.2.1.3

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

$l = \frac{92.48}{9.8696044}$

Step 4.2.2.1.4

Divide $92.48$ by $9.8696044$ .

$l = 9.37018306$