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:

• 4^{1 / 2} = 2 This can also be expressed as √4 = 2 2 is the square root of 4.
• 8^{1 / 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.