2.  Complex Numbers

3.  Square Root- For any real numbers a and b, if a 2 =b, then a is the square root of b.  Imaginary Unit- I, or the principal square root of -1, i 2 = -1.  Pure Imaginary Numbers- The square roots of negative real numbers, for any positive real number b, √-b 2 = √b 2 x √-1, or b x i.  Square Root Property- For any real number n, if x 2 = n, then x = ±√n.  Complex Number- Any number that can be written in the form a + bi, where a and b are real numbers and I is the imaginary unit.  Complex Conjugates- Two complex numbers in the form a + bi and a – bi.

4.  √ab = √a x √b  Example o √20 o √12 o √45

6.  √ a / b = √a / √b  Examples o √ 9 / 25 o √ 4 / 49

8.  i = √ -1  i 2 = -1  i 3 = -√ -1  i 4 = 1  i 5 = √ -1  i 6 = -1  i 7 = -√ -1  i 8 = 1 o Notice a pattern?

9.  Factor out i first  Then, break down all variables, if they are present, to the highest even power.  With numbers present, look to factor into perfect squares.  Simplify

10.  √-20  √-150x 7  √-32  √-252y 6

12.  When Given Square Roots o Factor out I, if necessary o Combine square roots o Factor into perfect squares o Simplify  When Given i o Multiply i values o Factor out even number of i’s o Multiply non-i values o Simplify

13.  -3i x 6i  √-3 x √-15  4i x -5i  √-6 x √-8

15.  Solve for variable  Don’t forget to use ±  Example o 4x 2 + 64 = 0 o 3x 2 + 75 = 0 o x 2 + 9 = 0

17.  Find values for the variables that make the equation true  Set corresponding parts equal to each other  Solve for the variables  Examples o 8 + 15i = 2m + 3ni o (r+1) +3si = 5 – 9i o (2x + 5) + (1 – n)i = -2 + 4i

19.  Combine values that do not have i in them  Combine values that do have i in them  Examples  (5-3i) + (2 + 4i)  (2 - i) – (6 - 5i)

21.  Multiply the top and the bottom by conjugates  Simplify  Examples

23.  Worksheet 5-4