Powers of
The imaginary unit is defined as the square root of . So, .
can be written as , which equals or simply .
can be written as , which equals or .
can be written as , which equals or .
Therefore, the cycle repeats every four powers, as shown in the table.
Powers of |
|
etc. | etc. |
Example 1:
Simplify.
Simplify the imaginary part using the property of multiplying powers.
Recall the definition of .
Since :
Example 2:
Simplify.
Rewrite the expression grouping the real and imaginary parts.
Simplify the imaginary part using the property of multiplying powers.
Recall the definition of .
Since :
Example 3:
Simplify the principal square roots.
Taking the square root and substituting :
Simplify.
Example 4:
Simplify the principal square roots.
Factor the radicand into squares,
Rewrite the expression using .
Note: When you are simplifying radicals as part of an equation, please remember that unless a principal square root is requested, there are always both positive and negative roots - including when you are working with .