Find the Magnitude || square root of 17-5i square root of 2||

$| | \sqrt{17} - 5 i \sqrt{2} | |$

Step 1

Reorder $- 5 i$ and $\sqrt{2}$ .

$| | \sqrt{17} + \sqrt{2} (- 5 i) | |$

Step 2

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

$| \sqrt{{\sqrt{17}}^{2} + {(\sqrt{2} \cdot - 5)}^{2}} |$

Step 3

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

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

$| \sqrt{{(17^{\frac{1}{2}})}^{2} + {(\sqrt{2} \cdot - 5)}^{2}} |$

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

$| \sqrt{17^{\frac{1}{2} \cdot 2} + {(\sqrt{2} \cdot - 5)}^{2}} |$

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

$| \sqrt{17^{\frac{2}{2}} + {(\sqrt{2} \cdot - 5)}^{2}} |$

Cancel the common factor of $2$ .

Cancel the common factor.

$| \sqrt{17^{\frac{2}{2}} + {(\sqrt{2} \cdot - 5)}^{2}} |$

Rewrite the expression.

$| \sqrt{17^{1} + {(\sqrt{2} \cdot - 5)}^{2}} |$

Evaluate the exponent.

$| \sqrt{17 + {(\sqrt{2} \cdot - 5)}^{2}} |$

Step 4

Move $- 5$ to the left of $\sqrt{2}$ .

$| \sqrt{17 + {(- 5 \cdot \sqrt{2})}^{2}} |$

Step 5

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

$| \sqrt{17 + {(- 5)}^{2} {\sqrt{2}}^{2}} |$

Step 6

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

$| \sqrt{17 + 25 {\sqrt{2}}^{2}} |$

Step 7

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

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

$| \sqrt{17 + 25 {(2^{\frac{1}{2}})}^{2}} |$

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

$| \sqrt{17 + 25 \cdot 2^{\frac{1}{2} \cdot 2}} |$

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

$| \sqrt{17 + 25 \cdot 2^{\frac{2}{2}}} |$

Cancel the common factor of $2$ .

Cancel the common factor.

$| \sqrt{17 + 25 \cdot 2^{\frac{2}{2}}} |$

Rewrite the expression.

$| \sqrt{17 + 25 \cdot 2^{1}} |$

Evaluate the exponent.

$| \sqrt{17 + 25 \cdot 2} |$

Step 8

Multiply $25$ by $2$ .

$| \sqrt{17 + 50} |$

Step 9

Add $17$ and $50$ .

$| \sqrt{67} |$

Step 10

$\sqrt{67}$ is approximately $8.18535277$ which is positive so remove the absolute value

$\sqrt{67}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\sqrt{67}$

Decimal Form:

$8.18535277 \dots$

Find the Magnitude || square root of 17-5i square root of 2||

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