Give three answers that are equivalent to the expression: i^ 1012

The answer can be i^4 , i^32 , i^124

Step-by-step explanation:

* Lets explain the the imaginary number i

- The number i is an imaginary number because its value is √(-1)

∵ i = √(-1)

∴ i² = -1

∴ i³ = -i

∴ i^4 = 1

∴ i^5 = i and so on

- Then if you want to know i^n = ?, you must to check n

# If n is an even number divisible by 4 then the answer is 1

# If n is an even number not divisible by 4 then the answer is -1

- Ex: i^16 = 1 but i^26 = -1

# If n is an odd number , then the answer will be i or - i, to know check

  n - 1 if it divisible by 4, then the answer is i if not divisible by 4, then

  the answer is - i

- Ex: i^13 = i , but i^15 = - i

* Now lets solve the problem

∵ The expression is i^1012

∵ 1012 is an even number and divisible by 4

∴  i^1012 = 1

∴ The answer can be i^4 , i^32 , i^124

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.

step-by-step explanation: