Question

question
simplify the expression to a + bi form:
$6i^{35} + 7i^{28} - 8i^{77} + 9i^{62}$

Explanation:

Step1: Simplify powers of $i$

Recall the cycle of $i$: $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$, and it repeats every 4.

• $i^{35}=i^{4\times8+3}=i^3=-i$
• $i^{28}=i^{4\times7}=i^4=1$
• $i^{77}=i^{4\times19+1}=i^1=i$
• $i^{62}=i^{4\times15+2}=i^2=-1$

Step2: Substitute back into expression

$6(-i) + 7(1) - 8(i) + 9(-1)$

Step3: Expand and combine like terms

$ -6i + 7 - 8i - 9 $
Combine real parts: $7-9=-2$
Combine imaginary parts: $-6i-8i=-14i$

Answer:

$-2 - 14i$