Until now, we have been dealing with real numbers. We have not been able to take the square root of a negative number because the square root of a negative number is not a real number. Instead, the square root of a negative number is an imaginary number--a number of the form , where k < 0. Imaginary numbers are represented as ki, where i = . For example, = 5i and = i.

We can simplify square roots of negative numbers by factoring out = i and simplifying the resulting root.

Examples:

1. Simplify .
   

   
   

   	=	·
   	=	i·
   	=	i·4·
   	=	4i.

   
   
2. Simplify .
   

   
   

   	=	·
   	=	i·10
   	=	10i.

   
   
3. Simplify .
   

   
   

   	=	·
   	=	i·
   	=	i·5·
   	=	5i.

   
   

Observe the following:

Thus, we can find iⁿ using the following:

• If n÷4 leaves a remainder of 1, iⁿ = i.
• If n÷4 leaves a remainder of 2, iⁿ = - 1.
• If n÷4 leaves a remainder of 3, iⁿ = - i.
• If n÷4 leaves no remainder, iⁿ = 1.

Examples:

1. What is i⁵⁴?
   54÷4 = 13R2.
   Thus, i⁵⁴ = - 1.
2. What is i¹⁰³?
   103÷4 = 25R3.
   Thus, i¹⁰³ = - i.