# 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

```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 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:

```(√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.

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