Algebra Examples

$t = 2 π \sqrt{\frac{L}{32}}$

Step 1

Simplify $2 π \sqrt{\frac{L}{32}}$ .

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Step 1.1

Rewrite $\frac{L}{32}$ as ${(\frac{1}{4})}^{2} \frac{L}{2}$ .

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Step 1.1.1

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

$t = 2 π \sqrt{\frac{1^{2} L}{32}}$

Step 1.1.2

Factor the perfect power $4^{2}$ out of $32$ .

$t = 2 π \sqrt{\frac{1^{2} L}{4^{2} \cdot 2}}$

Step 1.1.3

Rearrange the fraction $\frac{1^{2} L}{4^{2} \cdot 2}$ .

$t = 2 π \sqrt{{(\frac{1}{4})}^{2} \frac{L}{2}}$

Step 1.2

Pull terms out from under the radical.

$t = 2 π (\frac{1}{4} \sqrt{\frac{L}{2}})$

Step 1.3

Rewrite $\sqrt{\frac{L}{2}}$ as $\frac{\sqrt{L}}{\sqrt{2}}$ .

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L}}{\sqrt{2}})$

Step 1.4

Multiply $\frac{\sqrt{L}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = 2 π (\frac{1}{4} (\frac{\sqrt{L}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))$

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

Multiply $\frac{\sqrt{L}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})$

Step 1.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})$

Step 1.5.4

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

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

Step 1.5.5

Add $1$ and $1$ .

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 1.5.6

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

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Step 1.5.6.1

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

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 1.5.6.2

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

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 1.5.6.3

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

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.5.6.4

Cancel the common factor of $2$ .

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Step 1.5.6.4.1

Cancel the common factor.

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.5.6.4.2

Rewrite the expression.

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2^{1}})$

Step 1.5.6.5

Evaluate the exponent.

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L} \sqrt{2}}{2})$

Step 1.6

Combine using the product rule for radicals.

$t = 2 π (\frac{1}{4} \cdot \frac{\sqrt{L \cdot 2}}{2})$

Step 1.7

Multiply $\frac{1}{4} \cdot \frac{\sqrt{L \cdot 2}}{2}$ .

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Step 1.7.1

Multiply $\frac{1}{4}$ by $\frac{\sqrt{L \cdot 2}}{2}$ .

$t = 2 π \frac{\sqrt{L \cdot 2}}{4 \cdot 2}$

Step 1.7.2

Multiply $4$ by $2$ .

$t = 2 π \frac{\sqrt{L \cdot 2}}{8}$

Step 1.8

Cancel the common factor of $2$ .

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Step 1.8.1

Factor $2$ out of $2 π$ .

$t = 2 (π) \frac{\sqrt{L \cdot 2}}{8}$

Step 1.8.2

Factor $2$ out of $8$ .

$t = 2 (π) \frac{\sqrt{L \cdot 2}}{2 (4)}$

Step 1.8.3

Cancel the common factor.

$t = 2 π \frac{\sqrt{L \cdot 2}}{2 \cdot 4}$

Step 1.8.4

Rewrite the expression.

$t = π \frac{\sqrt{L \cdot 2}}{4}$

Step 1.9

Combine $π$ and $\frac{\sqrt{L \cdot 2}}{4}$ .

$t = \frac{π \sqrt{L \cdot 2}}{4}$

Step 1.10

Move $2$ to the left of $L$ .

$t = \frac{π \sqrt{2 L}}{4}$

Step 2

To rewrite as a function of $L$ , write the equation so that $t$ is by itself on one side of the equal sign and an expression involving only $L$ is on the other side.

$f (L) = \frac{π \sqrt{2 L}}{4}$