Simplify | 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

Simplify the expression.

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

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

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

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

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

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

Step 5

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 6

Simplify the expression.

Multiply $25$ by $2$ .

$\sqrt{17 + 50}$

Add $17$ and $50$ .

$\sqrt{67}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{67}$

Decimal Form:

$8.18535277 \dots$

Simplify | square root of 17-5i square root of 2|

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