Linear Algebra Examples

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

Step 1

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

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

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $4$ and $16$ .

$\sqrt{{\sqrt{20}}^{2} + {| 2 - 4 i |}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

Factor $4$ out of $20$ .

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

Step 2.5.2

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

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

Step 2.6

Pull terms out from under the radical.

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

Tap for more steps...

Step 2.9.1

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

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

Step 2.9.2

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

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

Step 2.9.3

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

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

Step 2.9.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.9.4.1

Cancel the common factor.

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

Step 2.9.4.2

Rewrite the expression.

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

Step 2.9.5

Evaluate the exponent.

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

Step 2.10

Multiply $4$ by $5$ .

$\sqrt{20 + {| 2 - 4 i |}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.11

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

$\sqrt{20 + {\sqrt{2^{2} + {(- 4)}^{2}}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.12

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

$\sqrt{20 + {\sqrt{4 + {(- 4)}^{2}}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.13

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

$\sqrt{20 + {\sqrt{4 + 16}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.14

Add $4$ and $16$ .

$\sqrt{20 + {\sqrt{20}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.15

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

Tap for more steps...

Step 2.15.1

Factor $4$ out of $20$ .

$\sqrt{20 + {\sqrt{4 (5)}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.15.2

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

$\sqrt{20 + {\sqrt{2^{2} \cdot 5}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.16

Pull terms out from under the radical.

$\sqrt{20 + {(2 \sqrt{5})}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.17

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

$\sqrt{20 + 2^{2} {\sqrt{5}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.18

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

$\sqrt{20 + 4 {\sqrt{5}}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.19

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

Tap for more steps...

Step 2.19.1

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

$\sqrt{20 + 4 {(5^{\frac{1}{2}})}^{2} + {| 1 - 2 i |}^{2}}$

Step 2.19.2

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

$\sqrt{20 + 4 \cdot 5^{\frac{1}{2} \cdot 2} + {| 1 - 2 i |}^{2}}$

Step 2.19.3

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

$\sqrt{20 + 4 \cdot 5^{\frac{2}{2}} + {| 1 - 2 i |}^{2}}$

Step 2.19.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.19.4.1

Cancel the common factor.

$\sqrt{20 + 4 \cdot 5^{\frac{2}{2}} + {| 1 - 2 i |}^{2}}$

Step 2.19.4.2

Rewrite the expression.

$\sqrt{20 + 4 \cdot 5^{1} + {| 1 - 2 i |}^{2}}$

Step 2.19.5

Evaluate the exponent.

$\sqrt{20 + 4 \cdot 5 + {| 1 - 2 i |}^{2}}$

Step 2.20

Multiply $4$ by $5$ .

$\sqrt{20 + 20 + {| 1 - 2 i |}^{2}}$

Step 2.21

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

$\sqrt{20 + 20 + {\sqrt{1^{2} + {(- 2)}^{2}}}^{2}}$

Step 2.22

One to any power is one.

$\sqrt{20 + 20 + {\sqrt{1 + {(- 2)}^{2}}}^{2}}$

Step 2.23

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

$\sqrt{20 + 20 + {\sqrt{1 + 4}}^{2}}$

Step 2.24

Add $1$ and $4$ .

$\sqrt{20 + 20 + {\sqrt{5}}^{2}}$

Step 2.25

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

Tap for more steps...

Step 2.25.1

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

$\sqrt{20 + 20 + {(5^{\frac{1}{2}})}^{2}}$

Step 2.25.2

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

$\sqrt{20 + 20 + 5^{\frac{1}{2} \cdot 2}}$

Step 2.25.3

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

$\sqrt{20 + 20 + 5^{\frac{2}{2}}}$

Step 2.25.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.25.4.1

Cancel the common factor.

$\sqrt{20 + 20 + 5^{\frac{2}{2}}}$

Step 2.25.4.2

Rewrite the expression.

$\sqrt{20 + 20 + 5^{1}}$

Step 2.25.5

Evaluate the exponent.

$\sqrt{20 + 20 + 5}$

Step 2.26

Add $20$ and $20$ .

$\sqrt{40 + 5}$

Step 2.27

Add $40$ and $5$ .

$\sqrt{45}$

Step 2.28

Rewrite $45$ as $3^{2} \cdot 5$ .

Tap for more steps...

Step 2.28.1

Factor $9$ out of $45$ .

$\sqrt{9 (5)}$

Step 2.28.2

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

$\sqrt{3^{2} \cdot 5}$

Step 2.29

Pull terms out from under the radical.

$3 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5}$

Decimal Form:

$6.70820393 \dots$