Trigonometry Examples

$\sqrt{\frac{586}{576}}$

Step 1

Cancel the common factor of $586$ and $576$ .

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Step 1.1

Factor $2$ out of $586$ .

$\sqrt{\frac{2 (293)}{576}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $576$ .

$\sqrt{\frac{2 \cdot 293}{2 \cdot 288}}$

Step 1.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 293}{2 \cdot 288}}$

Step 1.2.3

Rewrite the expression.

$\sqrt{\frac{293}{288}}$

Step 2

Rewrite $\sqrt{\frac{293}{288}}$ as $\frac{\sqrt{293}}{\sqrt{288}}$ .

$\frac{\sqrt{293}}{\sqrt{288}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 3.1.1

Factor $144$ out of $288$ .

$\frac{\sqrt{293}}{\sqrt{144 (2)}}$

Step 3.1.2

Rewrite $144$ as $12^{2}$ .

$\frac{\sqrt{293}}{\sqrt{12^{2} \cdot 2}}$

Step 3.2

Pull terms out from under the radical.

$\frac{\sqrt{293}}{12 \sqrt{2}}$

Step 4

Multiply $\frac{\sqrt{293}}{12 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{293}}{12 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt{293}}{12 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{293} \sqrt{2}}{12 \sqrt{2} \sqrt{2}}$

Step 5.2

Move $\sqrt{2}$ .

$\frac{\sqrt{293} \sqrt{2}}{12 (\sqrt{2} \sqrt{2})}$

Step 5.3

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

$\frac{\sqrt{293} \sqrt{2}}{12 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 5.4

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

$\frac{\sqrt{293} \sqrt{2}}{12 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 5.5

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

$\frac{\sqrt{293} \sqrt{2}}{12 {\sqrt{2}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$\frac{\sqrt{293} \sqrt{2}}{12 {\sqrt{2}}^{2}}$

Step 5.7

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

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Step 5.7.1

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

$\frac{\sqrt{293} \sqrt{2}}{12 {(2^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{\sqrt{293} \sqrt{2}}{12 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$\frac{\sqrt{293} \sqrt{2}}{12 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\frac{\sqrt{293} \sqrt{2}}{12 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$\frac{\sqrt{293} \sqrt{2}}{12 \cdot 2^{1}}$

Step 5.7.5

Evaluate the exponent.

$\frac{\sqrt{293} \sqrt{2}}{12 \cdot 2}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt{293 \cdot 2}}{12 \cdot 2}$

Step 6.2

Multiply $293$ by $2$ .

$\frac{\sqrt{586}}{12 \cdot 2}$

Step 7

Multiply $12$ by $2$ .

$\frac{\sqrt{586}}{24}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{586}}{24}$

Decimal Form:

$1.00864320 \dots$