To simplify [tex]\( 7i^{22} \)[/tex], let's begin by understanding the properties of the imaginary unit [tex]\( i \)[/tex].
The imaginary unit [tex]\( i \)[/tex] is defined by the property [tex]\( i^2 = -1 \)[/tex]. From this, we can derive the higher powers of [tex]\( i \)[/tex]:
[tex]\[
i^1 = i \\
i^2 = -1 \\
i^3 = i \cdot i^2 = i \cdot (-1) = -i \\
i^4 = (i^2)^2 = (-1)^2 = 1
\][/tex]
Notice that [tex]\( i^4 = 1 \)[/tex], and this pattern repeats every four powers:
[tex]\[
i^5 = i^1 = i \\
i^6 = i^2 = -1 \\
i^7 = i^3 = -i \\
i^8 = i^4 = 1
\][/tex]
This cyclical property of [tex]\( i \)[/tex] tells us that any integer power of [tex]\( i \)[/tex] can be reduced by considering the remainder when the exponent is divided by 4.
Next, we simplify [tex]\( i^{22} \)[/tex].
Find the remainder when 22 is divided by 4:
[tex]\[
22 \div 4 = 5 \text{ remainder } 2
\][/tex]
Therefore, [tex]\( i^{22} \)[/tex] can be simplified as:
[tex]\[
i^{22} = i^{4 \cdot 5 + 2} = (i^4)^5 \cdot i^2
\][/tex]
Since [tex]\( i^4 = 1 \)[/tex], we have:
[tex]\[
(i^4)^5 = 1^5 = 1
\][/tex]
Thus:
[tex]\[
i^{22} = 1 \cdot i^2 = i^2
\][/tex]
And since [tex]\( i^2 = -1 \)[/tex], we get:
[tex]\[
i^{22} = -1
\][/tex]
Now we need to multiply this result by 7:
[tex]\[
7i^{22} = 7 \cdot (-1) = -7
\][/tex]
Therefore, the simplified form of [tex]\( 7i^{22} \)[/tex] is [tex]\( -7 \)[/tex].
So the correct answer is:
[tex]\[ -7 \][/tex]
