Ms. Arrowood is incorrect in classifying the number \(\sqrt{-18}\) as a real number. The square root of a negative number is not a real number; instead, it is an imaginary number.
To express \(\sqrt{-18}\), we can break it down:
\[ \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} \]
Since \(\sqrt{-1}\) is defined as the imaginary unit \(i\), we have:
\[ \sqrt{-18} = \sqrt{18} \cdot i \]
We can simplify \(\sqrt{18}\):
\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
Therefore,
\[ \sqrt{-18} = 3\sqrt{2} \cdot i \]
In summary, \(\sqrt{-18}\) is an imaginary number (specifically \(3\sqrt{2}i\)), and therefore it is not a real number.
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