Sure! Let's match each of the given expressions with their correct answers based on what we know about powers of the imaginary unit [tex]\( j \)[/tex] (where [tex]\( j \)[/tex] represents [tex]\(\sqrt{-1}\)[/tex] or [tex]\( i \)[/tex]).
Given:
[tex]$j^7, j^8, j^4, j^3, j^6, j^2, j^5$[/tex]
And their possible values:
[tex]$1, -1, j, -j$[/tex]
### Step-by-Step Solution:
1. Determine [tex]\( j^7 \)[/tex]:
- [tex]\( j^7 = -j \)[/tex] or written as [tex]\( -i \)[/tex].
2. Determine [tex]\( j^8 \)[/tex]:
- [tex]\( j^8 = 1 \)[/tex].
3. Determine [tex]\( j^4 \)[/tex]:
- [tex]\( j^4 = 1 \)[/tex].
4. Determine [tex]\( j^3 \)[/tex]:
- [tex]\( j^3 = -j \)[/tex] or written as [tex]\( -i \)[/tex].
5. Determine [tex]\( j^6 \)[/tex]:
- [tex]\( j^6 = -1 \)[/tex].
6. Determine [tex]\( j^2 \)[/tex]:
- [tex]\( j^2 = -1 \)[/tex].
7. Determine [tex]\( j^5 \)[/tex]:
- [tex]\( j^5 = j \)[/tex] or written as [tex]\( i \)[/tex].
Now, we straightforwardly match these results with the given expressions:
[tex]$\underline{\space} \ j^7 \quad (-j)$[/tex]
[tex]$j^8 \quad (1)$[/tex]
[tex]$j^4 \quad (1)$[/tex]
[tex]$j^3 \quad (-j)$[/tex]
[tex]$j^6 \quad (-1)$[/tex]
[tex]$\underline{\space} \ j^2 \quad (-1)$[/tex]
[tex]$\underline{\space} \ j^5 \quad (j)$[/tex]
### Let's fill them into the original expressions correctly:
1. [tex]\( j^7 = -j \)[/tex]
- The box before [tex]\( j^7 \)[/tex]:
[tex]$-\boxed{j}$[/tex]
2. [tex]\( j^8 = 1 \)[/tex]
3. [tex]\( j^4 = 1 \)[/tex]
4. [tex]\( j^3 = -j \)[/tex]
- The box before [tex]\( j^3 \)[/tex]:
[tex]$-\boxed{j}$[/tex]
5. [tex]\( j^6 = -1 \)[/tex]
6. [tex]\( j^2 = -1 \)[/tex]
- The box before [tex]\( j^2 \)[/tex]:
[tex]$\boxed{-1}$[/tex]
7. [tex]\( j^5 = j \)[/tex]
- The box before [tex]\( j^5 \)[/tex]:
[tex]$\boxed{j}$[/tex]
Hence, plugging these into their boxes:
[tex]\[
\begin{align*}
&(-j) \quad j^8 \quad j^4 \quad (-j) \quad j^6 \quad (-1) \quad j^5 \\
\end{align*}
\][/tex]
Now I am going to summarize it more neatly:
[tex]$ -j \ j^7 $[/tex]
[tex]$ j^8 = 1 $[/tex]
[tex]$ j^4 = 1 $[/tex]
[tex]$ -j \ j^3 $[/tex]
[tex]$ j^6 = -1 $[/tex]
[tex]$ -1 \ j^2 $[/tex]
[tex]$ j^5 $[/tex]
