Linear Algebra Examples

$[\begin{array}{c} 2 - 2 i \\ 4 + 4 i \\ 4 - 1 i \end{array}]$

Step 1

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

$\sqrt{{| 2 - 2 i |}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$\sqrt{{\sqrt{2^{2} + {(- 2)}^{2}}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.2

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

$\sqrt{{\sqrt{4 + {(- 2)}^{2}}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.3

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

$\sqrt{{\sqrt{4 + 4}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.4

Add $4$ and $4$ .

$\sqrt{{\sqrt{8}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.5

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 2.5.1

Factor $4$ out of $8$ .

$\sqrt{{\sqrt{4 (2)}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.5.2

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

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

Step 2.6

Pull terms out from under the radical.

$\sqrt{{(2 \sqrt{2})}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.7

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

$\sqrt{2^{2} {\sqrt{2}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.8

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

$\sqrt{4 {\sqrt{2}}^{2} + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.9

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

Tap for more steps...

Step 2.9.1

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

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

Step 2.9.2

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

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

Step 2.9.3

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

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

Step 2.9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.9.4.1

Cancel the common factor.

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

Step 2.9.4.2

Rewrite the expression.

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

Step 2.9.5

Evaluate the exponent.

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

Step 2.10

Multiply $4$ by $2$ .

$\sqrt{8 + {| 4 + 4 i |}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.11

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

$\sqrt{8 + {\sqrt{4^{2} + 4^{2}}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.12

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

$\sqrt{8 + {\sqrt{16 + 4^{2}}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.13

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

$\sqrt{8 + {\sqrt{16 + 16}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.14

Add $16$ and $16$ .

$\sqrt{8 + {\sqrt{32}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.15

Rewrite $32$ as $4^{2} \cdot 2$ .

Tap for more steps...

Step 2.15.1

Factor $16$ out of $32$ .

$\sqrt{8 + {\sqrt{16 (2)}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.15.2

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

$\sqrt{8 + {\sqrt{4^{2} \cdot 2}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.16

Pull terms out from under the radical.

$\sqrt{8 + {(4 \sqrt{2})}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.17

Apply the product rule to $4 \sqrt{2}$ .

$\sqrt{8 + 4^{2} {\sqrt{2}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.18

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

$\sqrt{8 + 16 {\sqrt{2}}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.19

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

Tap for more steps...

Step 2.19.1

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

$\sqrt{8 + 16 {(2^{\frac{1}{2}})}^{2} + {| 4 - 1 i |}^{2}}$

Step 2.19.2

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

$\sqrt{8 + 16 \cdot 2^{\frac{1}{2} \cdot 2} + {| 4 - 1 i |}^{2}}$

Step 2.19.3

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

$\sqrt{8 + 16 \cdot 2^{\frac{2}{2}} + {| 4 - 1 i |}^{2}}$

Step 2.19.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.19.4.1

Cancel the common factor.

$\sqrt{8 + 16 \cdot 2^{\frac{2}{2}} + {| 4 - 1 i |}^{2}}$

Step 2.19.4.2

Rewrite the expression.

$\sqrt{8 + 16 \cdot 2^{1} + {| 4 - 1 i |}^{2}}$

Step 2.19.5

Evaluate the exponent.

$\sqrt{8 + 16 \cdot 2 + {| 4 - 1 i |}^{2}}$

Step 2.20

Multiply $16$ by $2$ .

$\sqrt{8 + 32 + {| 4 - 1 i |}^{2}}$

Step 2.21

Rewrite $- 1 i$ as $- i$ .

$\sqrt{8 + 32 + {| 4 - i |}^{2}}$

Step 2.22

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

$\sqrt{8 + 32 + {\sqrt{4^{2} + {(- 1)}^{2}}}^{2}}$

Step 2.23

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

$\sqrt{8 + 32 + {\sqrt{16 + {(- 1)}^{2}}}^{2}}$

Step 2.24

Raise $- 1$ to the power of $2$ .

$\sqrt{8 + 32 + {\sqrt{16 + 1}}^{2}}$

Step 2.25

Add $16$ and $1$ .

$\sqrt{8 + 32 + {\sqrt{17}}^{2}}$

Step 2.26

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

Tap for more steps...

Step 2.26.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\sqrt{8 + 32 + {(17^{\frac{1}{2}})}^{2}}$

Step 2.26.2

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

$\sqrt{8 + 32 + 17^{\frac{1}{2} \cdot 2}}$

Step 2.26.3

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

$\sqrt{8 + 32 + 17^{\frac{2}{2}}}$

Step 2.26.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.26.4.1

Cancel the common factor.

$\sqrt{8 + 32 + 17^{\frac{2}{2}}}$

Step 2.26.4.2

Rewrite the expression.

$\sqrt{8 + 32 + 17^{1}}$

Step 2.26.5

Evaluate the exponent.

$\sqrt{8 + 32 + 17}$

Step 2.27

Add $8$ and $32$ .

$\sqrt{40 + 17}$

Step 2.28

Add $40$ and $17$ .

$\sqrt{57}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{57}$

Decimal Form:

$7.54983443 \dots$