Question

Evaluate the power, and write the result in the form $a+bi$.

$i^{1000}$

Short answer

The value of power is $1+0i$.

Step-by-step solution

Step 1. Concept of complex number.

A complex number is of the form $a+bi$. Where a and b are some real numbers and i is the imaginary number.

The number a is known as the real part of the complex number and b is known as imaginary part of the complex number.

Step 2. Concept of imaginary number.

The imaginary number i is defined as the square root of negative one. That is, $i^2=-1$.

The exponent properties are defined as below,

• $a^m\cdot a^n=a^{m+n}$
  
• ${\left(ab\right)}^m=a^m\cdot b^m$
  
• ${\left(a^m\right)}^n=a^{m\times n}$
  

Step 3. Find the power.

To evaluate the given power, rewrite the power as a product or exponent of$i^2$ and simplify further using the exponent properties as shown below

role="math" localid="1644267046366" $$\begin{gathered} i^{1000}={\left(i^2\right)}^{500} \\ ={\left(-1\right)}^{500} \\ =1 \\ =1+0i \end{gathered}$$

Thus, the value of the power is $1+0i$.