Algebra Examples

$t = 2 π \sqrt{\frac{l}{384}}$

Step 1

Rewrite the equation as $2 π \sqrt{\frac{l}{384}} = t$ .

$2 π \sqrt{\frac{l}{384}} = t$

Step 2

Simplify $2 π \sqrt{\frac{l}{384}}$ .

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Step 2.1

Rewrite $\frac{l}{384}$ as ${(\frac{1}{8})}^{2} \frac{l}{6}$ .

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Step 2.1.1

Factor the perfect power $1^{2}$ out of $l$ .

$2 π \sqrt{\frac{1^{2} l}{384}} = t$

Step 2.1.2

Factor the perfect power $8^{2}$ out of $384$ .

$2 π \sqrt{\frac{1^{2} l}{8^{2} \cdot 6}} = t$

Step 2.1.3

Rearrange the fraction $\frac{1^{2} l}{8^{2} \cdot 6}$ .

$2 π \sqrt{{(\frac{1}{8})}^{2} \frac{l}{6}} = t$

Step 2.2

Pull terms out from under the radical.

$2 π (\frac{1}{8} \sqrt{\frac{l}{6}}) = t$

Step 2.3

Rewrite $\sqrt{\frac{l}{6}}$ as $\frac{\sqrt{l}}{\sqrt{6}}$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l}}{\sqrt{6}}) = t$

Step 2.4

Multiply $\frac{\sqrt{l}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$2 π (\frac{1}{8} (\frac{\sqrt{l}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})) = t$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{\sqrt{l}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{\sqrt{6} \sqrt{6}}) = t$

Step 2.5.2

Raise $\sqrt{6}$ to the power of $1$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}) = t$

Step 2.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}) = t$

Step 2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}) = t$

Step 2.5.5

Add $1$ and $1$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{{\sqrt{6}}^{2}}) = t$

Step 2.5.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}) = t$

Step 2.5.6.2

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

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}) = t$

Step 2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{6^{\frac{2}{2}}}) = t$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{6^{\frac{2}{2}}}) = t$

Step 2.5.6.4.2

Rewrite the expression.

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{6^{1}}) = t$

Step 2.5.6.5

Evaluate the exponent.

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l} \sqrt{6}}{6}) = t$

Step 2.6

Combine using the product rule for radicals.

$2 π (\frac{1}{8} \cdot \frac{\sqrt{l \cdot 6}}{6}) = t$

Step 2.7

Multiply $\frac{1}{8} \cdot \frac{\sqrt{l \cdot 6}}{6}$ .

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Step 2.7.1

Multiply $\frac{1}{8}$ by $\frac{\sqrt{l \cdot 6}}{6}$ .

$2 π \frac{\sqrt{l \cdot 6}}{8 \cdot 6} = t$

Step 2.7.2

Multiply $8$ by $6$ .

$2 π \frac{\sqrt{l \cdot 6}}{48} = t$

Step 2.8

Cancel the common factor of $2$ .

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Step 2.8.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{\sqrt{l \cdot 6}}{48} = t$

Step 2.8.2

Factor $2$ out of $48$ .

$2 (π) \frac{\sqrt{l \cdot 6}}{2 (24)} = t$

Step 2.8.3

Cancel the common factor.

$2 π \frac{\sqrt{l \cdot 6}}{2 \cdot 24} = t$

Step 2.8.4

Rewrite the expression.

$π \frac{\sqrt{l \cdot 6}}{24} = t$

Step 2.9

Combine $π$ and $\frac{\sqrt{l \cdot 6}}{24}$ .

$\frac{π \sqrt{l \cdot 6}}{24} = t$

Step 2.10

Move $6$ to the left of $l$ .

$\frac{π \sqrt{6 l}}{24} = t$

Step 3

Subtract $\frac{π \sqrt{6 l}}{24}$ from both sides of the equation.

$0 = t - \frac{π \sqrt{6 l}}{24}$

Step 4

Rewrite the equation as $t - \frac{π \sqrt{6 l}}{24} = 0$ .

$t - \frac{π \sqrt{6 l}}{24} = 0$