1. Find the values of the following.
1) (1 + i)^16
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1
Answer:
Given: A complex number 1+ i
To find: (1+i)^{16}(1+i)
16
Step-by-step explanation:
As given
(1+i)^{16}(1+i)
16
can be written as
\begin{gathered}((1+i)^2)^8 \text {---} [\because (a^m)^n+a^{mn}]\\\\\end{gathered}
((1+i)
2
)
8
—[∵(a
m
)
n
+a
mn
]
Now as we know
(a+b)^2= a^2+b^2+2ab(a+b)
2
=a
2
+b
2
+2ab we have
(1^1+i^2+2\times 1\times i)^8(1
1
+i
2
+2×1×i)
8
Now i^2= -1i
2
=−1 we get
\begin{gathered}(1-1+2i)^8\\\\=(2i)^8=2^8\times (i^2)^4= 256 \times (-1)^4= 256\times 1 =256\end{gathered}
(1−1+2i)
8
=(2i)
8
=2
8
×(i
2
)
4
=256×(−1)
4
=256×1=256
Hence the value of (1+i)^{16}(1+i)
16
is 256
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