Algebra Examples

$6 = 2 π \sqrt{\frac{L + 4}{9.8}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{L + 4}{9.8}} = 6$ .

$2 π \sqrt{\frac{L + 4}{9.8}} = 6$

Step 2

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

${(2 π \sqrt{\frac{L + 4}{9.8}})}^{2} = 6^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{L + 4}{9.8}}$ as ${(\frac{L + 4}{9.8})}^{\frac{1}{2}}$ .

${(2 π {(\frac{L + 4}{9.8})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(2 π {(\frac{L + 4}{9.8})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply by $1$ .

${(2 π {(\frac{1 L + 4}{9.8})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.2

Rewrite $4$ as $1 (4)$ .

${(2 π {(\frac{1 L + 1 (4)}{9.8})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.3

Factor $1$ out of $1 L + 1 (4)$ .

${(2 π {(\frac{1 (L + 4)}{9.8})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.4

Factor $9.8$ out of $9.8$ .

${(2 π {(\frac{1 (L + 4)}{9.8 (1)})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.5

Separate fractions.

${(2 π {(\frac{1}{9.8} \cdot \frac{L + 4}{1})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.6

Simplify the expression.

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Step 3.2.1.6.1

Divide $1$ by $9.8$ .

${(2 π {(0.10204081 \frac{L + 4}{1})}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.6.2

Divide $L + 4$ by $1$ .

${(2 π {(0.10204081 (L + 4))}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.7

Apply the distributive property.

${(2 π {(0.10204081 L + 0.10204081 \cdot 4)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.8

Multiply $0.10204081$ by $4$ .

${(2 π {(0.10204081 L + 0.40816326)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.9

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

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Step 3.2.1.9.1

Apply the product rule to $2 π {(0.10204081 L + 0.40816326)}^{\frac{1}{2}}$ .

${(2 π)}^{2} {({(0.10204081 L + 0.40816326)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.9.2

Apply the product rule to $2 π$ .

$2^{2} π^{2} {({(0.10204081 L + 0.40816326)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.10

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

$4 π^{2} {({(0.10204081 L + 0.40816326)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 3.2.1.11

Multiply the exponents in ${({(0.10204081 L + 0.40816326)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.11.1

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

$4 π^{2} {(0.10204081 L + 0.40816326)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.2.1.11.2

Cancel the common factor of $2$ .

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Step 3.2.1.11.2.1

Cancel the common factor.

$4 π^{2} {(0.10204081 L + 0.40816326)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 3.2.1.11.2.2

Rewrite the expression.

$4 π^{2} {(0.10204081 L + 0.40816326)}^{1} = 6^{2}$

Step 3.2.1.12

Simplify.

$4 π^{2} (0.10204081 L + 0.40816326) = 6^{2}$

Step 3.2.1.13

Apply the distributive property.

$4 π^{2} (0.10204081 L) + 4 π^{2} \cdot 0.40816326 = 6^{2}$

Step 3.2.1.14

Multiply $4 π^{2} (0.10204081 L)$ .

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Step 3.2.1.14.1

Multiply $0.10204081$ by $4$ .

$0.40816326 π^{2} L + 4 π^{2} \cdot 0.40816326 = 6^{2}$

Step 3.2.1.14.2

Multiply $0.40816326$ by $π^{2}$ .

$4.02840995 L + 4 π^{2} \cdot 0.40816326 = 6^{2}$

Step 3.2.1.15

Multiply $4 π^{2} \cdot 0.40816326$ .

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Step 3.2.1.15.1

Multiply $0.40816326$ by $4$ .

$4.02840995 L + 1.63265306 π^{2} = 6^{2}$

Step 3.2.1.15.2

Multiply $1.63265306$ by $π^{2}$ .

$4.02840995 L + 16.11363983 = 6^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $6$ to the power of $2$ .

$4.02840995 L + 16.11363983 = 36$

Step 4

Solve for $L$ .

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Step 4.1

Move all terms not containing $L$ to the right side of the equation.

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Step 4.1.1

Subtract $16.11363983$ from both sides of the equation.

$4.02840995 L = 36 - 16.11363983$

Step 4.1.2

Subtract $16.11363983$ from $36$ .

$4.02840995 L = 19.88636016$

Step 4.2

Divide each term in $4.02840995 L = 19.88636016$ by $4.02840995$ and simplify.

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Step 4.2.1

Divide each term in $4.02840995 L = 19.88636016$ by $4.02840995$ .

$\frac{4.02840995 L}{4.02840995} = \frac{19.88636016}{4.02840995}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4.02840995$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4.02840995 L}{4.02840995} = \frac{19.88636016}{4.02840995}$

Step 4.2.2.1.2

Divide $L$ by $1$ .

$L = \frac{19.88636016}{4.02840995}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $19.88636016$ by $4.02840995$ .

$L = 4.93652839$