To express \( 5 - 2 - \sqrt{18} \) as a complex number in the standard form \( a + bi \), let's first simplify the given expression step by step:
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Simplify \( \sqrt{18} \): \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
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Now, substitute this back into the expression: \[ 5 - 2 - \sqrt{18} = 5 - 2 - 3\sqrt{2} \]
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Combine the real parts: \[ 5 - 2 = 3 \]
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Thus, the expression simplifies to: \[ 3 - 3\sqrt{2} \]
Now we need to determine which of the provided responses matches our result. Note that the expression as it stands is purely real, and does not have an imaginary component \( bi \).
Looking at the options provided:
- \( 5 - 3i \sqrt{2} \)
- \( 5 - 6i \sqrt{2} \)
- \( 5 - 6 \sqrt{2} \)
- \( 5 - 3\sqrt{2}i \)
None of these options directly reflect our simplification of \( 3 - 3\sqrt{2} \), as they all contain the imaginary unit \( i \).
However, if we look closely, if we consider the option:
- \( 5 - 3 \sqrt{2} i \) resembles our needed simplification. Thus our final answer must be expressed differently or altered based on any additional details in the choices given.
Since no choice matches \( 3 - 3\sqrt{2} \), the conclusion can be drawn that none fulfill the requirement as intended to stand fully in \( a + bi \).
However, if looking strictly for a complex representation that narrows towards the form, among the complex options, \( 5 - 3 \sqrt{2} i \) is the suitable choice despite differing \( a - bi \) structure.
Please verify if my calculated component or the selections require editing or reflecting true assumptions leading towards your examination directives or clarifications.
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