Question

question
simplify the expression to a + bi form:
$8i^{119} + i^{78} - 3i^{97} + 9i^{44}$

Explanation:

Step1: Use $i$ cycle property

Recall $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$; cycle length 4.

Step2: Reduce exponents mod 4

• $119 \div 4 = 29$ rem $3$, so $i^{119}=i^3=-i$
• $78 \div 4 = 19$ rem $2$, so $i^{78}=i^2=-1$
• $97 \div 4 = 24$ rem $1$, so $i^{97}=i^1=i$
• $44 \div 4 = 11$ rem $0$, so $i^{44}=i^4=1$

Step3: Substitute values into expression

$8(-i) + (-1) - 3(i) + 9(1)$

Step4: Simplify the expression

$=-8i -1 -3i +9$
$=(-1+9) + (-8i-3i)$
$=8 -11i$

Answer:

$8 - 11i$