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Imaginary Numbers Explained with Formula

$$ \sqrt{-1} \text{= ?} $$

What they are and how to simplify

Part I. (this Page)
Simplifying Powers of $$i$$
$$ i^5 = ? \\ i ^ {21} = ? $$
Part II. Simplifying negative radicands
$$ \sqrt{-25} = ? \\ \sqrt{-18} = ? $$

What is an imaginary number anyway?

question
The General Formula

$$ i^k = $$question is the same as $$ i^ \color{Red}{r}$$ where $$ \color{Red}{r} $$ is the remainder when k is divided by 4 .
Whether the remainder is 1 , 2 , 3 , or 4 , the key to simplifying powers of i is the remainder when the exponent is divided by 4 .
simplifying the powers of the imaginary number i
Practice Problems


Problem 1) $$ i^ {23} $$
Answer


Problem 2) $$ i^ {18} $$
Answer


Problem 3) $$ i^ {41} $$
Answer


Problem 4) $$ i^ {100} $$
Answer


Problem 5) $$ 5i^ {22} $$
Answer


Problem 6) $$ 12i^ {36} $$
Answer


Problem 7) $$ 7 i^ {103} $$
Answer