Question

compute ( i^{-138} ).

Explanation:

Step1: Rewrite negative exponent

$i^{-138} = \frac{1}{i^{138}}$

Step2: Simplify $i^{138}$ via cycle

Since $i^4=1$, divide 138 by 4: $138 = 4\times34 + 2$, so $i^{138}=(i^4)^{34}\times i^2 = 1^{34}\times i^2 = i^2$

Step3: Substitute $i^2=-1$

$\frac{1}{i^{138}} = \frac{1}{-1} = -1$

Answer:

$-1$