Alan

Quadrant IV

The x (real) value is positive, the y (imaginary) value is negative.

"how to simplify 7^3*7^-2/7^5"

Use the fact that  a^b*a^c = a^(b+c)  and  a^b/a^c = a^(b-c)

\(\frac{7^3\times7^{-2}}{7^5}\rightarrow\frac{7^1}{7^5}\rightarrow7^{-4}\rightarrow \frac{1}{7^4}\)

.

As follows:

.

A.  Remainder = 3

1.  n = 3:  (n^2)/5 = 1 rem 4;  (n^3)/5 = 5 rem 2;  n/5 = 0 rem 3

B.  3, 5, 7

Probability first card is a heart: p1 = 13/52

Provability that second card is a clover (club): p2 = 13/51

p1 and p2 = (13/52)*(13/51) → 13/204

.

Electric field:

.

"A 6-sided die is weighted so that the probability of any number being rolled is proportional to the value of the roll. (So, for example, the probability of a 2 being rolled is twice that of a 1 being rolled.) What is the expected value of a roll of this weighted die?"

E = (1*1 + 2*2 + 3*3 + 4*4 + 5*5 + 6*6)/(1 + 2 + 3 + 4 + 5 + 6)

E =  91/21 

E = 13/3

.

mod(1009^1,101) = 100

mod(1009^2,101) = 1

mod(1009^3,101) = 100

mod(1009^4,101) = 1

mod(1009^5,101) = 100

...

mod(1009^109,101) = 100

.

This graph might help:

.

Or as follows:

.