Complex Numbers And Quadratic Equations question 682
Question: What is the value of $ {{( -\sqrt{-1} )}^{4n+3}}+{{( i^{41}+{i^{-257}} )}^{9}}, $ where $ n\in N $ ?
Options:
A) 0
B) 1
C) i
D) $ -i $
Show Answer
Answer:
Correct Answer: C
Solution:
Consider $ {{(-\sqrt{-1})}^{4n+3}}+{{(i^{41}-{i^{-257}})}^{9}} $ $ ={{(-i)}^{4n+3}}+{{[ {{(i^{4})}^{10}}.i^{1}+{{(i^{3})}^{-85}}.{i^{-2}} ]}^{9}} $ $ ={{(-i)}^{4n+3}}+{{[ i+\frac{1}{{{(i^{3})}^{85}}}.\frac{1}{i^{2}} ]}^{9}} $ $ ={{(-i)}^{4n+3}}+{{( i+\frac{1}{i} )}^{9}} $ $ =-{{(-1)}^{4n+3}}{{(i)}^{4n}}{{(i)}^{3}}+{{(i-i)}^{9}}=-(1)(-i)+0=i $
BETA
