Question:
If $z = i^9 + i^{19}$ , then $z$ is equal to
If $z = i^9 + i^{19}$ , then $z$ is equal to
Updated On: Apr 8, 2024
- 0 + 0i
- 1 + 0i
- 0 + i
- 1 + 2 i
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The Correct Option is A
Solution and Explanation
We have
$ z =i^{9}+i^{19} $
$=\left(i^{4}\right)^{2} \cdot i+\left(i^{4}\right)^{4} \cdot i^{3}$
$=i+i^{3}\,\,\, \left[\because i^{4}=1\right]$
$=i-i \,\,\,\left[\because i^{3}=-i\right]$
$=0$
$=0+0 i $
$ z =i^{9}+i^{19} $
$=\left(i^{4}\right)^{2} \cdot i+\left(i^{4}\right)^{4} \cdot i^{3}$
$=i+i^{3}\,\,\, \left[\because i^{4}=1\right]$
$=i-i \,\,\,\left[\because i^{3}=-i\right]$
$=0$
$=0+0 i $
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
