Please assist me with the power of i(imaginary)

Let's raise i to various powers starting with 0,1,2,3...

i^0 = 1

i^1 = i

i^2 = ( sqrt(-1) )^2 = -1

i^3 = i^2*i = -1*i = -i

i^4 = (i^2)^2 = (-1)^2 = 1

i^5 = i^4*i = 1*i = i

i^6 = i^5*i = i*i = i^2 = -1

We see that the pattern repeats itself after 4 iterations. The four items to memorize are

i^0 = 1

i^1 = i

i^2 = -1

i^3 = -i

It bounces back and forth between 1 and i, alternating in sign as well. This could be one way to memorize the pattern.

To figure out something like i^25, we simply divide the exponent 25 over 4 to get the remainder. In this case, the remainder of 25/4 is 1 since 24/4 = 6, and 25 is one higher than 24.

This means i^25 = i^1 = i

Likewise,

i^5689 = i^1 = i

because 5689/4 = 1422 remainder 1. The quotient doesn't play a role at all so you can ignore it entirely

simplifying

5(x + -6) = 2(x + 3)

reorder the terms:

5(-6 + x) = 2(x + 3)

(-6 * 5 + x * 5) = 2(x + 3)

(-30 + 5x) = 2(x + 3)

reorder the terms:

-30 + 5x = 2(3 + x)

-30 + 5x = (3 * 2 + x * 2)

-30 + 5x = (6 + 2x)

solving

-30 + 5x = 6 + 2x

solving for variable 'x'.

move all terms containing x to the left, all other terms to the right.

add '-2x' to each side of the equation.

-30 + 5x + -2x = 6 + 2x + -2x

combine like terms: 5x + -2x = 3x

-30 + 3x = 6 + 2x + -2x

combine like terms: 2x + -2x = 0

-30 + 3x = 6 + 0

-30 + 3x = 6

add '30' to each side of the equation.

-30 + 30 + 3x = 6 + 30

combine like terms: -30 + 30 = 0

0 + 3x = 6 + 30

3x = 6 + 30

combine like terms: 6 + 30 = 36

3x = 36

divide each side by '3'.

x = 12

simplifying

x = 12

this should .