Trigonometry Examples

$\sqrt{\sqrt{\frac{18}{121}}}$

Step 1

Rewrite $\sqrt{\frac{18}{121}}$ as $\frac{\sqrt{18}}{\sqrt{121}}$ .

$\sqrt{\frac{\sqrt{18}}{\sqrt{121}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

$\sqrt{\frac{\sqrt{9 (2)}}{\sqrt{121}}}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{\sqrt{3^{2} \cdot 2}}{\sqrt{121}}}$

Step 2.2

Pull terms out from under the radical.

$\sqrt{\frac{3 \sqrt{2}}{\sqrt{121}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $121$ as $11^{2}$ .

$\sqrt{\frac{3 \sqrt{2}}{\sqrt{11^{2}}}}$

Step 3.2

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\frac{3 \sqrt{2}}{11}}$

Step 4

Rewrite $\sqrt{\frac{3 \sqrt{2}}{11}}$ as $\frac{\sqrt{3 \sqrt{2}}}{\sqrt{11}}$ .

$\frac{\sqrt{3 \sqrt{2}}}{\sqrt{11}}$

Step 5

Multiply $\frac{\sqrt{3 \sqrt{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{3 \sqrt{2}}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{3 \sqrt{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 6.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 6.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 6.4

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

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 6.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{11^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{11^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{\sqrt{3 \sqrt{2}} \sqrt{11}}{11}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \sqrt{2} \cdot 11}}{11}$

Step 7.2

Multiply $11$ by $3$ .

$\frac{\sqrt{33 \sqrt{2}}}{11}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{33 \sqrt{2}}}{11}$

Decimal Form:

$0.62104316 \dots$