We need to calculate
(1−4i)3. We can do this by expanding the expression using the binomial theorem or by multiplying it out directly. Let's use the direct multiplication method:
(1−4i)3=(1−4i)(1−4i)(1−4i) First, we multiply the first two factors:
(1−4i)(1−4i)=1−4i−4i+16i2Remember that
i2=−1, so we can simplify:
1−4i−4i+16i2=1−8i−16=−15−8i
Now, we multiply this result by the remaining factor (1-4i):
(−15−8i)(1−4i)=−15+60i−8i+32i2
Simplifying again using
i2=−1 :
−15+60i−8i+32i2=−15+52i−32=−47+52i
Therefore,
(1−4i)3=−47+52i. Comparing this with the form
a+bi, we find that
a=−47 and
b=52.
So the correct answer is Option A: -47, 52.