GRE Arithmetic | Exponents and Roots

A number can be expressed in terms of base and its powers or exponent or index (Number = Base^exponent). The base of a number can be a positive or negative integer fraction or decimal. Let a number 4, it can be expressed as 2 * 2 or in terms of base and exponent 2². Here base of the number is 2 and exponent is also 2.
Example

8 = 23
125 = 53
1296 = 362
1024 = 210
1000 = 103

Possible operations on Exponents:

xa * xb = xa + b
(xa)b = xa*b
xa / xb = xa - b
(xy)a = xayb
(x)-a = (1 / x)a
(x / y)a = xa / yb

Examples on operations:

 22*27 = 29 = 512
 (52)3 = 55 = 3125
 37 / 34 = 34 = 81
 (3 * 5)2 = 32 * 52 225
 5-1 = (1 / 5)1 = 1 / 5 = 0.2
 (3 / 4)5 = 35 / 45 = 243 / 1024
 

When a number are in form of Base^{1 / exponent} then it yields the roots of the number. Root of a number can be square root, cube root and
For example:

 41 / 2 = 2
This can also be expressed as √4 = 2
2 is the square root of 4.
81 / 3 = 2
This can also be expressed as ∛8 = 2
2 is the cube root of 8

Simple operation on roots:

(√x)2 = x
√(x2) = x
√x * √y = √(x*y) 
√x / √y = √(x / y) 
√x + √y = √x + √y ≠ √(x + y) 
√x - √y = √x - √y ≠ √(x - y)  

For example:

(√5)2 = 5
√(52) = 5
√5 * √3 = √(5*3) =  √(15) 
√10 / √3 = √(10 / 3) 
√5 + √3 = √5 + √3 ≠ √8 
√5 - √3 = √5 - √3 ≠ √2  

Note: Root of a negative number can be expressed in terms of imaginary number.