To classify each expression as either a real number or a complex number, let us examine them individually:
1. [tex]\(\sqrt{(-3)^2}\)[/tex]:
- Calculate [tex]\( (-3)^2 \)[/tex]: [tex]\( (-3)^2 = 9 \)[/tex].
- Then, take the square root of 9: [tex]\( \sqrt{9} = 3 \)[/tex].
- This is a real number.
2. [tex]\( -2 + 5i \)[/tex]:
- This expression has both a real part ([tex]\(-2\)[/tex]) and an imaginary part ([tex]\(5i\)[/tex]).
- This is a complex number.
3. [tex]\( i^4 \)[/tex]:
- Considering the powers of [tex]\( i \)[/tex]:
- [tex]\( i^2 = -1 \)[/tex]
- [tex]\( i^4 = (i^2)^2 = (-1)^2 = 1 \)[/tex]
- This is a real number.
4. [tex]\( \sqrt{-3} \)[/tex]:
- The square root of a negative number involves [tex]\( i \)[/tex].
- Specifically, [tex]\( \sqrt{-3} = \sqrt{3}i \)[/tex].
- This is a complex number.
5. [tex]\( 7i^2 + 6i^3 \)[/tex]:
- Calculate [tex]\( i^2 \)[/tex]: [tex]\( i^2 = -1 \)[/tex].
- Calculate [tex]\( i^3 \)[/tex]: [tex]\( i^3 = i^2 \cdot i = -i \)[/tex].
- So, [tex]\( 7i^2 = 7(-1) = -7 \)[/tex].
- And, [tex]\( 6i^3 = 6(-i) = -6i \)[/tex].
- Combining: [tex]\( 7i^2 + 6i^3 = -7 - 6i \)[/tex].
- This is a complex number.
6. [tex]\( 0 + 4i \)[/tex]:
- This expression resembles the form [tex]\( a + bi \)[/tex] with [tex]\( a = 0 \)[/tex] and [tex]\( b = 4 \)[/tex].
- This is a complex number.
7. [tex]\( -1 + 3i^2 \)[/tex]:
- Considering [tex]\( i^2 = -1 \)[/tex]:
- Calculate [tex]\( 3i^2 \)[/tex]: [tex]\( 3 \cdot (-1) = -3 \)[/tex].
- So, the expression becomes [tex]\( -1 + (-3) = -4 \)[/tex].
- This is a real number.
8. 409:
- This is a plain real number (an integer).
- This is a real number.
Now, organizing the expressions into their respective categories:
[tex]\[
\begin{tabular}{|l|l|}
\hline Real Numbers & Complex Numbers \\
\hline
& \\
\sqrt{(-3)^2} & -2 + 5i \\
i^4 & \sqrt{-3} \\
-1 + 3i^2 & 7i^2 + 6i^3 \\
409 & 0 + 4i \\
& \\
\hline
\end{tabular}
\][/tex]
So, the final table should be:
[tex]\[
\begin{tabular}{|l|l|}
\hline Real Numbers & Complex Numbers \\
\hline
\sqrt{(-3)^2} & -2 + 5i \\
i^4 & \sqrt{-3} \\
-1 + 3i^2 & 7i^2 + 6i^3 \\
409 & 0 + 4i \\
\hline
\end{tabular}
\][/tex]
