Can anyone please explain me exponentiation.

Answer 1

Exponenting is simply just multiplicating a number by itself certain number of times.

5 ** 4 = 5 * 5 * 5 * 5 = 625
2 ** 3 = 2 * 2 * 2 = 8
8 ** 2 = 8 * 8 = 64
4 ** 1 = 4

Raising a value to a positive integer is relatively simple, but exponentation rules might get more complex when raising with negative values, decimal numbers or 0 can be more complicated.

Raising to negative integers will divide 1 with the exponented value, when the exponent is positive, hard to explain but here:
5 ** (-4) = 1 / (5 ** 4) = 1 / 625 = 0.0016
2 ** (-3) = 1 / (2 ** 3) = 1 / 8 = 0.125
8 ** (-2) = 1 / (8 ** 2) = 1 / 64 = 0.015625
4 ** (-1) = 1 / (4 ** 1) = 1 / 4 = 0.25

Raising to a decimal can give you a root value of a value, hard to tell why, but using power 1/2 you can get square root, with power 1/3 you can get cube root, with 1/100 you can get 100th root.
16 ** (1/2) = 4
25 ** (1/2) = 5
27 ** (1/3) = 3
256 ** (1/8) = 2

You can raise power with mixed decimals and integers:
4 ** (2 + 1/2) = 4 ** 2 * 4 ** (1/2) = 16 * 2 = 32

Answer 2

Raising with 0 is assumed to be 1.
5 ** 0 = 1
2 ** 0 = 1
8 ** 0 = 1
4 ** 0 = 1
0 ** 0 = 1
-1 ** 0 = 1

One of the most confusing part of exponenting is raising a negative number with half, because it's same than taking squareroot of negative numbers:
(-1)**(1/2) = "complex"
(-27)**(1/3) = -3
(-64)**(1/4) = "complex"
(-5)**(1/6) = (-5)**(1/2) * (-5)**(1/3) = "complex"

Like name describes it's complex, I'm not really sure where to use those, but they can be interesting and they exist in some programming languages such as Python.

Answer 3

2**5 == 2*2*2*2*2
7**3 == 7*7*7
4**4 == 4*4*4*4