Bot GPT 4-o mini To re-express \( 3 - \sqrt{32} \) as a complex number with the imaginary unit \( i \), we can first simplify the square root of 32.
We can rewrite \( \sqrt{32} \) as:
\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]
Now substitute this back into the expression:
\[ 3 - \sqrt{32} = 3 - 4\sqrt{2} \]
Next, we express this in terms of \( i \). The imaginary unit \( i \) is defined as:
\[ i = \sqrt{-1} \]
To relate \( 3 - 4\sqrt{2} \) to the imaginary unit, we can represent it in a complex form. Any real number can be expressed as a complex number with its imaginary part set to zero. Thus we write:
\[ 3 - 4\sqrt{2} = (3 - 4\sqrt{2}) + 0i \]
Therefore, we fully express \( 3 - \sqrt{32} \) as a complex number:
\[ 3 - \sqrt{32} = (3 - 4\sqrt{2}) + 0i \]
This is the expression of \( 3 - \sqrt{32} \) as a complex number where the imaginary part is \( 0 \).
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