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, = 5*i* and = *i*.

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

*Examples*:

- Simplify .
= · = *i*·= *i*·4·= 4 *i*. - Simplify .
= · = *i*·10= 10 *i*. - Simplify .
= · = *i*·= *i*·5·= 5 *i*.

Observe the following:

i^{1} | = | i | |

i^{2} | = | ()^{2} = - 1 | |

i^{3} | = | i^{2}i = - 1(i) = - i | |

i^{4} | = | i^{3}i = - i(i) = - i^{2} = - (- 1) = 1 | |

i^{5} | = | i^{4}i = 1(i) = i | |

i^{6} | = | i^{5}i = - 1 | |

i^{7} | = | i^{6}i = - i | |

i^{8} | = | i^{7}i = 1 | |

i^{9} | = | i | |

^{ ... } |

Thus, we can find

- If
*n*÷4 leaves a remainder of 1,*i*^{n}=*i*. - If
*n*÷4 leaves a remainder of 2,*i*^{n}= - 1. - If
*n*÷4 leaves a remainder of 3,*i*^{n}= -*i*. - If
*n*÷4 leaves no remainder,*i*^{n}= 1.

*Examples*:

- What is
*i*^{54}?

54÷4 = 13*R*2.

Thus,*i*^{54}= - 1. - What is
*i*^{103}?

103÷4 = 25*R*3.

Thus,*i*^{103}= -*i*.

Take a Study Break!