GRE Arithmetic | Exponents and Roots

A number can be expressed in terms of base and its powers or exponent or index (Number = Baseexponent). 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

  1. 8 = 23
  2. 125 = 53
  3. 1296 = 362
  4. 1024 = 210
  5. 1000 = 103

Possible operations on Exponents:

  1. xa * xb = xa + b
  2. (xa)b = xa*b
  3. xa / xb = xa - b
  4. (xy)a = xayb
  5. (x)-a = (1 / x)a
  6. (x / y)a = xa / yb

 
Examples on operations:

  1. 22*27 = 29 = 512
  2. (52)3 = 55 = 3125
  3. 37 / 34 = 34 = 81
  4. (3 * 5)2 = 32 * 52 225
  5. 5-1 = (1 / 5)1 = 1 / 5 = 0.2
  6. (3 / 4)5 = 35 / 45 = 243 / 1024

When a number are in form of Base1 / 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:

  1. (√x)2 = x
  2. √(x2) = x
  3. √x * √y = √(x*y)
  4. √x / √y = √(x / y)
  5. √x + √y = √x + √y ≠ √(x + y)
  6. √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.



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