Question

If \[z = {i^9} + {i^{19}}\], then \[z\]is equal to
A. \[0 + 0i\]
B. \[1 + 0i\]
C. \[0 + i\]
D. \[1 + 2i\]

Answer 1

Hint: Here we will use the property of powers of complex numbers, that is \[{i^n}\] equals to \[i\] when \[n\]is one more than the multiple of 4 and \[{i^n}\] is equal to \[ - i\] when \[n\]is three more than the multiple of 4.
* A complex number \[z = x + iy\] has real part \[x\] and imaginary part \[y\].
* Since \[i = \sqrt { - 1} \] therefore, \[{i^2} = - 1,{i^3} = i \times {i^2} = i \times ( - 1) = - i,{i^4} = {({i^2})^2} = {( - 1)^2} = 1\]
* We break the power if \[i\] is multiple of \[4\] because \[{i^4} = 1\] and the rest powers can be allotted to another \[i\] .
* Also, in addition or subtraction of two complex numbers, always add the real part to the real part and the imaginary part to the imaginary part of the complex number.
* Here we use the concept of the same base but different powers i.e. we have the same base \[i\] therefore, powers can be separated, added or subtracted.

Complete step by step solution:
Given, \[z = {i^9} + {i^{19}}\]
Since, \[9 = 4 \times 2 + 1\]
\[{i^9}\]can be written as \[{i^{4 \times 2 + 1}}\]. Therefore from the property of powers of complex number when exponent of \[i\]is one more than multiple of \[4\]
i.e. \[{i^9} = {i^{4 \times 2 + 1}} = {({i^2})^4} \times (i) = 1 \times i = i\] \[...(1)\]
Now, \[19 = 4 \times 4 + 3\]
Similarly \[{i^{19}}\] can be written as \[{i^{4 \times 4 + 3}}\]. Therefore from the property of powers of complex number when exponent of \[i\]is three more than multiple of 4
i.e. \[{i^{19}} = {i^{4 \times 4 + 3}} = {({i^4})^4} \times ({i^3}) = {1^4} \times ( - i) = - i\] \[...(2)\]
Substituting the values of \[{i^9}\] and \[{i^{19}}\] from equations \[(1)\] and \[(2)\] \[z\] can be simplified to\[z = {i^9} + {i^{19}} = i + ( - i) = i - i = 0\]
Write the obtained value of \[z\] in the form of \[a + bi\].

Therefore, \[z = 0 + 0i\].

Therefore, option (A) is the correct answer.

Note:
In these types of questions where power of \[i\] is involved, the exponent should be simplified in terms of multiples of \[4\], as \[{i^{4n}} = 1\]. Also all the multiples of \[4\] can directly be written equal to \[1\]. Whenever we get \[0\] as the answer always write the answer in form of a complex number that indicates both the real and the imaginary part as \[0\] i.e. \[z = 0 + 0i\]