Algebra Examples

$t = 2 π \sqrt{\frac{1}{32}}$

Step 1

Rewrite $\sqrt{\frac{1}{32}}$ as $\frac{\sqrt{1}}{\sqrt{32}}$ .

$t = 2 π \frac{\sqrt{1}}{\sqrt{32}}$

Step 2

Any root of $1$ is $1$ .

$t = 2 π \frac{1}{\sqrt{32}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.1.1

Factor $16$ out of $32$ .

$t = 2 π \frac{1}{\sqrt{16 (2)}}$

Step 3.1.2

Rewrite $16$ as $4^{2}$ .

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

Step 3.2

Pull terms out from under the radical.

$t = 2 π \frac{1}{4 \sqrt{2}}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $2 π$ .

$t = 2 (π) \frac{1}{4 \sqrt{2}}$

Step 4.1.2

Factor $2$ out of $4 \sqrt{2}$ .

$t = 2 (π) \frac{1}{2 (2 \sqrt{2})}$

Step 4.1.3

Cancel the common factor.

$t = 2 π \frac{1}{2 (2 \sqrt{2})}$

Step 4.1.4

Rewrite the expression.

$t = π \frac{1}{2 \sqrt{2}}$

Step 4.2

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

$t = \frac{π}{2 \sqrt{2}}$

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

$t = \frac{π \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 6.2

Move $\sqrt{2}$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

$t = \frac{π \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$t = \frac{π \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

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

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

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

Step 6.7.4.2

Rewrite the expression.

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

Step 6.7.5

Evaluate the exponent.

$t = \frac{π \sqrt{2}}{2 \cdot 2}$

Step 7

Multiply $2$ by $2$ .

$t = \frac{π \sqrt{2}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = \frac{π \sqrt{2}}{4}$

Decimal Form:

$t = 1.11072073 \dots$