Evaluate t=2pi square root of l/32

$t = 2 π \sqrt{\frac{l}{32}}$

Step 1

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

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

$t = 2 π \sqrt{\frac{1^{2} l}{32}}$

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

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

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

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

Step 2

Pull terms out from under the radical.

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

Step 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 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 5

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

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

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

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

Multiply $4$ by $2$ .

$t = 2 π \frac{\sqrt{l \cdot 2}}{8}$

Step 8

Cancel the common factor of $2$ .

Factor $2$ out of $2 π$ .

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

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$t = π \frac{\sqrt{l \cdot 2}}{4}$

Step 9

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

$t = \frac{π \sqrt{l \cdot 2}}{4}$

Step 10

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

$t = \frac{π \sqrt{2 l}}{4}$

Evaluate t=2pi square root of l/32

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