Linear Algebra Examples

$[\begin{array}{c} 4 + 3 i \\ 4 - 2 i \\ 0 - 4 i \end{array}]$

Step 1

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{{| 4 + 3 i |}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2

Simplify.

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Step 2.1

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

$\sqrt{{\sqrt{4^{2} + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.2

Raise $4$ to the power of $2$ .

$\sqrt{{\sqrt{16 + 3^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.3

Raise $3$ to the power of $2$ .

$\sqrt{{\sqrt{16 + 9}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.4

Add $16$ and $9$ .

$\sqrt{{\sqrt{25}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.5

Rewrite $25$ as $5^{2}$ .

$\sqrt{{\sqrt{5^{2}}}^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.6

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

$\sqrt{5^{2} + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.7

Raise $5$ to the power of $2$ .

$\sqrt{25 + {| 4 - 2 i |}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.8

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

$\sqrt{25 + {\sqrt{4^{2} + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.9

Raise $4$ to the power of $2$ .

$\sqrt{25 + {\sqrt{16 + {(- 2)}^{2}}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.10

Raise $- 2$ to the power of $2$ .

$\sqrt{25 + {\sqrt{16 + 4}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.11

Add $16$ and $4$ .

$\sqrt{25 + {\sqrt{20}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.12

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.12.1

Factor $4$ out of $20$ .

$\sqrt{25 + {\sqrt{4 (5)}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.12.2

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

$\sqrt{25 + {\sqrt{2^{2} \cdot 5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.13

Pull terms out from under the radical.

$\sqrt{25 + {(2 \sqrt{5})}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.14

Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{25 + 2^{2} {\sqrt{5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.15

Raise $2$ to the power of $2$ .

$\sqrt{25 + 4 {\sqrt{5}}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.16

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

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Step 2.16.1

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

$\sqrt{25 + 4 {(5^{\frac{1}{2}})}^{2} + {| 0 - 4 i |}^{2}}$

Step 2.16.2

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

$\sqrt{25 + 4 \cdot 5^{\frac{1}{2} \cdot 2} + {| 0 - 4 i |}^{2}}$

Step 2.16.3

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

$\sqrt{25 + 4 \cdot 5^{\frac{2}{2}} + {| 0 - 4 i |}^{2}}$

Step 2.16.4

Cancel the common factor of $2$ .

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Step 2.16.4.1

Cancel the common factor.

$\sqrt{25 + 4 \cdot 5^{\frac{2}{2}} + {| 0 - 4 i |}^{2}}$

Step 2.16.4.2

Rewrite the expression.

$\sqrt{25 + 4 \cdot 5^{1} + {| 0 - 4 i |}^{2}}$

Step 2.16.5

Evaluate the exponent.

$\sqrt{25 + 4 \cdot 5 + {| 0 - 4 i |}^{2}}$

Step 2.17

Multiply $4$ by $5$ .

$\sqrt{25 + 20 + {| 0 - 4 i |}^{2}}$

Step 2.18

Subtract $4 i$ from $0$ .

$\sqrt{25 + 20 + {| - 4 i |}^{2}}$

Step 2.19

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

$\sqrt{25 + 20 + {\sqrt{0^{2} + {(- 4)}^{2}}}^{2}}$

Step 2.20

Raising $0$ to any positive power yields $0$ .

$\sqrt{25 + 20 + {\sqrt{0 + {(- 4)}^{2}}}^{2}}$

Step 2.21

Raise $- 4$ to the power of $2$ .

$\sqrt{25 + 20 + {\sqrt{0 + 16}}^{2}}$

Step 2.22

Add $0$ and $16$ .

$\sqrt{25 + 20 + {\sqrt{16}}^{2}}$

Step 2.23

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

$\sqrt{25 + 20 + {\sqrt{4^{2}}}^{2}}$

Step 2.24

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

$\sqrt{25 + 20 + 4^{2}}$

Step 2.25

Raise $4$ to the power of $2$ .

$\sqrt{25 + 20 + 16}$

Step 2.26

Add $25$ and $20$ .

$\sqrt{45 + 16}$

Step 2.27

Add $45$ and $16$ .

$\sqrt{61}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{61}$

Decimal Form:

$7.81024967 \dots$