Linear Algebra Examples

$\sqrt{(\frac{1}{3}) {(0.0001)}^{2} + (\frac{1}{3}) {(- 0.0005)}^{2} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 1

Raise $0.0001$ to the power of $2$ .

$\sqrt{\frac{1}{3} \cdot 0.00000001 + (\frac{1}{3}) {(- 0.0005)}^{2} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 2

Cancel the common factor of $0.00000001$ .

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Step 2.1

Factor $0.00000001$ out of $3$ .

$\sqrt{\frac{1}{0.00000001 (300000000)} \cdot 0.00000001 + (\frac{1}{3}) {(- 0.0005)}^{2} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 2.2

Cancel the common factor.

$\sqrt{\frac{1}{0.00000001 \cdot 300000000} \cdot 0.00000001 + (\frac{1}{3}) {(- 0.0005)}^{2} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 2.3

Rewrite the expression.

$\sqrt{\frac{1}{300000000} + (\frac{1}{3}) {(- 0.0005)}^{2} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 3

Raise $- 0.0005$ to the power of $2$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{3} \cdot 0.00000025 + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 4

Cancel the common factor of $0.00000025$ .

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Step 4.1

Factor $0.00000025$ out of $3$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{0.00000025 (12000000)} \cdot 0.00000025 + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 4.2

Cancel the common factor.

$\sqrt{\frac{1}{300000000} + \frac{1}{0.00000025 \cdot 12000000} \cdot 0.00000025 + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 4.3

Rewrite the expression.

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + (\frac{1}{3}) {(0.0006)}^{2} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 5

Raise $0.0006$ to the power of $2$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + \frac{1}{3} \cdot 0.00000036 + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 6

Combine $\frac{1}{3}$ and $0.00000036$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + \frac{0.00000036}{3} + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 7

Simplify the expression.

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Step 7.1

Divide $0.00000036$ by $3$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + 0.00000012 + (\frac{1}{3}) {(- 0.0001)}^{2}}$

Step 7.2

Raise $- 0.0001$ to the power of $2$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + 0.00000012 + \frac{1}{3} \cdot 0.00000001}$

Step 8

Cancel the common factor of $0.00000001$ .

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Step 8.1

Factor $0.00000001$ out of $3$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + 0.00000012 + \frac{1}{0.00000001 (300000000)} \cdot 0.00000001}$

Step 8.2

Cancel the common factor.

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + 0.00000012 + \frac{1}{0.00000001 \cdot 300000000} \cdot 0.00000001}$

Step 8.3

Rewrite the expression.

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} + 0.00000012 + \frac{1}{300000000}}$

Step 9

To write $\frac{1}{12000000}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\sqrt{\frac{1}{300000000} + \frac{1}{12000000} \cdot \frac{25}{25} + 0.00000012 + \frac{1}{300000000}}$

Step 10

Write each expression with a common denominator of $300000000$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{1}{12000000}$ by $\frac{25}{25}$ .

$\sqrt{\frac{1}{300000000} + \frac{25}{12000000 \cdot 25} + 0.00000012 + \frac{1}{300000000}}$

Step 10.2

Multiply $12000000$ by $25$ .

$\sqrt{\frac{1}{300000000} + \frac{25}{300000000} + 0.00000012 + \frac{1}{300000000}}$

Step 11

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 25}{300000000} + 0.00000012 + \frac{1}{300000000}}$

Step 12

Add $1$ and $25$ .

$\sqrt{\frac{26}{300000000} + 0.00000012 + \frac{1}{300000000}}$

Step 13

To write $0.00000012$ as a fraction with a common denominator, multiply by $\frac{300000000}{300000000}$ .

$\sqrt{\frac{26}{300000000} + 0.00000012 \cdot \frac{300000000}{300000000} + \frac{1}{300000000}}$

Step 14

Combine $0.00000012$ and $\frac{300000000}{300000000}$ .

$\sqrt{\frac{26}{300000000} + \frac{0.00000012 \cdot 300000000}{300000000} + \frac{1}{300000000}}$

Step 15

Combine the numerators over the common denominator.

$\sqrt{\frac{26 + 0.00000012 \cdot 300000000}{300000000} + \frac{1}{300000000}}$

Step 16

Simplify the numerator.

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Step 16.1

Multiply $0.00000012$ by $300000000$ .

$\sqrt{\frac{26 + 36}{300000000} + \frac{1}{300000000}}$

Step 16.2

Add $26$ and $36$ .

$\sqrt{\frac{62}{300000000} + \frac{1}{300000000}}$

Step 17

Simplify the expression.

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Step 17.1

Combine the numerators over the common denominator.

$\sqrt{\frac{62 + 1}{300000000}}$

Step 17.2

Add $62$ and $1$ .

$\sqrt{\frac{63}{300000000}}$

Step 18

Cancel the common factor of $63$ and $300000000$ .

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Step 18.1

Factor $3$ out of $63$ .

$\sqrt{\frac{3 (21)}{300000000}}$

Step 18.2

Cancel the common factors.

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Step 18.2.1

Factor $3$ out of $300000000$ .

$\sqrt{\frac{3 \cdot 21}{3 \cdot 100000000}}$

Step 18.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 21}{3 \cdot 100000000}}$

Step 18.2.3

Rewrite the expression.

$\sqrt{\frac{21}{100000000}}$

Step 19

Rewrite $\sqrt{\frac{21}{100000000}}$ as $\frac{\sqrt{21}}{\sqrt{100000000}}$ .

$\frac{\sqrt{21}}{\sqrt{100000000}}$

Step 20

Simplify the denominator.

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Step 20.1

Rewrite $100000000$ as $10000^{2}$ .

$\frac{\sqrt{21}}{\sqrt{10000^{2}}}$

Step 20.2

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

$\frac{\sqrt{21}}{10000}$

Step 21

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{21}}{10000}$

Decimal Form:

$0.00045825 \dots$