To solve for the least positive and greatest negative integer values of k in the expression (‌
1−i
1+i
)k=−i, follow these steps: Start with the given equation: (‌
1−i
1+i
)k=−i Simplify the fraction ‌
1−i
1+i
: ‌
1−i
1+i
⋅‌
1−i
1−i
=‌
(1−i)2
(1+i)(1−i)
Calculate the numerator and the denominator: ‌(1−i)2=1−2i+i2=1−2i−1=−2i ‌(1+i)(1−i)=1+i−i−i2=1+1=2 Substitute back into the fraction: ‌
(1−i)2
(1+i)(1−i)
=‌
−2i
2
=−i Therefore, the expression simplifies to: (−i)k=−i For (−i)k=−i, it implies: k≡1(bmod‌4) From this, determine the least positive integer m and greatest negative integer n : m=1‌ (since ‌k=1‌ is the smallest positive integer solution) ‌ n=−3 (since k=−3 satisfies the condition −i in the sequence and is the largest negative solution) Calculate m−n : m−n=1−(−3)=4 Thus, m−n=4.