Hi Angelina! For these simultaneous equations, we use elimination by division. I will write up the workings before 11:30 pm if no one has already done them by then.

@Eric Nicholas K sure thank you very much! Really appreciate your help throughout this year!

y = ka^x

From the coordinates, we have :

when x = 2, y = 16
when x = 3, y = 32
when x = 5, y = p

we can form 3 equations by subbing these values.

From the first set of coordinates,
16 = ka² ①

From the second set,
32 = ka³ ②

From the third set,
p = ka^5 ③


since a and k are constants,

we can divide ② by ①,

32/16 = ka³/ka²

2 = a

(If it's a variable x then we cannot divide like this since we would be removing a possible result)


Now we sub a = 2 back into ②(or ① , up to you)

32 = k(2³) = 8k

k = 32/8 = 4


Now sub a = 2 and k = 4 into ③

p = 4(2^5) = 128