Express the following in the form a+i b $\begin{array}{ll}
\text { (i) } \frac{5+\sqrt{2} i}{1-\sqrt{2} i} & \text { (ii) } i^{-35}
\end{array}
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Step-by-Step Solution
Step 1: Simplify the first expression (i)
To express a complex number in the form a+ib, when it is given as a fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of
1−2i
is
1+2i
.
Step 2: Expand the numerator and denominator
We expand both the numerator and the denominator. In the denominator, we use the identity
(x−y)(x+y)=x2−y2
. Remember that
i2=−1
.
Step 3: Simplify the expression
Substitute
i2=−1
into the expression. Combine the real and imaginary terms in the numerator and simplify the denominator.
Step 4: Final form for (i)
Divide each term in the numerator by the denominator to get the expression in the form a+ib. Here,
a=1
and
b=22
.
Step 5: Simplify the second expression (ii)
For the second expression, we first rewrite
i−35
as
i351
.
Step 6: Simplify the power of i
We know that the powers of
i
repeat in a cycle of 4:
i1=i
,
i2=−1
,
i3=−i
,
i4=1
. To simplify
i35
, we divide the exponent by 4 and use the remainder.
35=4×8+3
, so
i35=i3=−i
.
Step 7: Substitute and rationalize
Substitute
i35=−i
back into the expression. To remove
i
from the denominator, multiply the numerator and denominator by
i
. Remember
i2=−1
.
Step 8: Final form for (ii)
The expression
i
can be written in the form a+ib as
0+1i
. Here,
a=0
and
b=1
.