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Patterns of i

 The Math standard covered is: "Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Common Core Math Standard: HSN-CN.A.2  

​Key point #1:

Patterns of i

Simplifying i to any power is easy!!!  
1. Take the exponent and divide by 4. (why 4? Because the pattern of i contains four values)
2. Look only at the remainder!
3. If the remainder is one then the value is i, if the remainder is 2, then the value is -1, if the remainder is 3, then the value is –i, if there is no remainder, the value is 1.

Key point #2:

Simplifying negative radical expressions

Ex. 1 √(-9)=√(-1) x√9=ix3 or 3i

Ex. 2 4i√(-72)=4i√(-1) x√72=4i^2 x6√2=4x6x(-1)x√2=-24√2

​Key point #3

Multiplying with i

(4- 2i)(6+5i) Use the FOIL method
4x6 + 4x5i – 2ix6 - 2ix5i =

24 + 20i - 12i - 10i2

24 + 8i – 10(-1)

24+10 + 8i = 34 + 8i (write in the a + bi form)

Video Guide
0:08 Video begins reviewing the four values of "i"
This includes -i, -i squared,-i cubed,and -i the the fourth
This creates a pattern of four that can be used to find the value of exponents.

1:56 This problem explains how to use the pattern of "i" You divide the exponents by four and then see what remainder is. 

2:05 Find the value of "i" for i^103

2:31 Find the value of "i" for i^39

3:00 Use the pattern of "i" to find the value of "i" for i^638

Example problems simplifying binomials that contain "i"

0:15 Problem 1 Multiply 3i times 6+5i 
This problem contains a monomial times a binomial and is just a distributive problem.

1:05 This problem is a binomial times a binomial.In order to solve this type of problem you just foil the problem.

2:30 This problem contains division.The tricky part of division problems is that you cannot leave an "i" in the denominator because "i" is the negative square root of 1.

4:23 This problem is another division problem, The way you handle the "i" in the denominator is by multiplying the fraction by "-i"/"-i"

Directions for solving radicals that have a negative value. This involves simplifying the negative radical. Sounds difficult,but after working a few problems you will find they aren't that hard.

Video Guide
0:24 5i times the square root of -64 You will solve this problem by simplifying the square root of 64. Next you convert the square root of -1 into "i" move the coefficient down and multiply.

1:48 Simplify square root of -81 + 3i -2 Again you simplify the square root, bring down the coefficient, and add like terms. Remember these problems should be written in the a + bi form.

3:00 This is another problem containing a fraction and a negative number.

4:04 The last problem simplifies the negative square root of 9 times the negative square root of 36

Algebra-Imaginary Numbers-Patterns of "i"

Algebra-Complex Numbers-Patterns of "i"

Algebra-Imaginary Numbers-Simplifying negative radicals