Trigonometry Examples

$\sqrt{{(\sqrt{11})}^{2 - {(\sqrt{22})}^{2}}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$\sqrt{{\sqrt{11}}^{2 - {(22^{\frac{1}{2}})}^{2}}}$

Step 1.1.2

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

$\sqrt{{\sqrt{11}}^{2 - 22^{\frac{1}{2} \cdot 2}}}$

Step 1.1.3

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

$\sqrt{{\sqrt{11}}^{2 - 22^{\frac{2}{2}}}}$

Step 1.1.4

Cancel the common factor of $2$ .

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Step 1.1.4.1

Cancel the common factor.

$\sqrt{{\sqrt{11}}^{2 - 22^{\frac{2}{2}}}}$

Step 1.1.4.2

Rewrite the expression.

$\sqrt{{\sqrt{11}}^{2 - 22^{1}}}$

Step 1.1.5

Evaluate the exponent.

$\sqrt{{\sqrt{11}}^{2 - 1 \cdot 22}}$

Step 1.2

Multiply $- 1$ by $22$ .

$\sqrt{{\sqrt{11}}^{2 - 22}}$

Step 2

Subtract $22$ from $2$ .

$\sqrt{{\sqrt{11}}^{- 20}}$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt{\frac{1}{{\sqrt{11}}^{20}}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite ${\sqrt{11}}^{20}$ as $11^{10}$ .

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Step 4.1.1

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

$\sqrt{\frac{1}{{(11^{\frac{1}{2}})}^{20}}}$

Step 4.1.2

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

$\sqrt{\frac{1}{11^{\frac{1}{2} \cdot 20}}}$

Step 4.1.3

Combine $\frac{1}{2}$ and $20$ .

$\sqrt{\frac{1}{11^{\frac{20}{2}}}}$

Step 4.1.4

Cancel the common factor of $20$ and $2$ .

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Step 4.1.4.1

Factor $2$ out of $20$ .

$\sqrt{\frac{1}{11^{\frac{2 \cdot 10}{2}}}}$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $2$ out of $2$ .

$\sqrt{\frac{1}{11^{\frac{2 \cdot 10}{2 (1)}}}}$

Step 4.1.4.2.2

Cancel the common factor.

$\sqrt{\frac{1}{11^{\frac{2 \cdot 10}{2 \cdot 1}}}}$

Step 4.1.4.2.3

Rewrite the expression.

$\sqrt{\frac{1}{11^{\frac{10}{1}}}}$

Step 4.1.4.2.4

Divide $10$ by $1$ .

$\sqrt{\frac{1}{11^{10}}}$

Step 4.2

Raise $11$ to the power of $10$ .

$\sqrt{\frac{1}{25937424601}}$

Step 5

Rewrite $\sqrt{\frac{1}{25937424601}}$ as $\frac{\sqrt{1}}{\sqrt{25937424601}}$ .

$\frac{\sqrt{1}}{\sqrt{25937424601}}$

Step 6

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{25937424601}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $25937424601$ as $161051^{2}$ .

$\frac{1}{\sqrt{161051^{2}}}$

Step 7.2

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

$\frac{1}{161051}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{161051}$

Decimal Form:

$6.20921323 \cdot 10^{- 6}$