Question

Re-express 3−32−−−−√
as a complex number with the imaginary unit i=−1−−−√
.(

Answer 1

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 \( 0 \).