Finite Math Examples

$| \sqrt{11} + i \sqrt{5} |$

Step 1

Reorder $i$ and $\sqrt{5}$ .

$| \sqrt{11} + \sqrt{5} i |$

Step 2

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{{\sqrt{11}}^{2} + {\sqrt{5}}^{2}}$

Step 3

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

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Step 3.1

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

$\sqrt{{(11^{\frac{1}{2}})}^{2} + {\sqrt{5}}^{2}}$

Step 3.2

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

$\sqrt{11^{\frac{1}{2} \cdot 2} + {\sqrt{5}}^{2}}$

Step 3.3

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

$\sqrt{11^{\frac{2}{2}} + {\sqrt{5}}^{2}}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

$\sqrt{11^{\frac{2}{2}} + {\sqrt{5}}^{2}}$

Step 3.4.2

Rewrite the expression.

$\sqrt{11^{1} + {\sqrt{5}}^{2}}$

Step 3.5

Evaluate the exponent.

$\sqrt{11 + {\sqrt{5}}^{2}}$

Step 4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{11 + {(5^{\frac{1}{2}})}^{2}}$

Step 4.2

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

$\sqrt{11 + 5^{\frac{1}{2} \cdot 2}}$

Step 4.3

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

$\sqrt{11 + 5^{\frac{2}{2}}}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\sqrt{11 + 5^{\frac{2}{2}}}$

Step 4.4.2

Rewrite the expression.

$\sqrt{11 + 5^{1}}$

Step 4.5

Evaluate the exponent.

$\sqrt{11 + 5}$

Step 5

Simplify the expression.

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Step 5.1

Add $11$ and $5$ .

$\sqrt{16}$

Step 5.2

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

$\sqrt{4^{2}}$

Step 5.3

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

$4$