# Vocabulary/hat

 ^ y Exponential

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

ey, the y-th power of the mathematical constant: e .

The antilogarithm of a natural logarithm.

```   ^1
2.71828
^2
7.38906
^ ^. 123.45
123.45
```

Logarithm (^. y)

### Use These Combinations

Combinations using ^ y that have exceptionally good performance include:

 What it does Type; Precisions; Ranks Syntax Variants; Restrictions Benefits; Bug Warnings eπy ^@o. y handles large values of y

 x ^ y Power

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

xy, the y-th power of numeric noun: x .

```   2^8
256
```

### Related Primitives

Logarithm (x ^. y)

### Details

1. Some results are defined in J that are often left undefined:

• 0 ^ 0 = 1
• _ ^ 0 = 1
• 1 ^ _ = 1

2. Taking a rational power of an extended integer produces a floating-point result whenever the denominator of the power is not 1.

### Use These Combinations

Combinations using x ^ y that have exceptionally good performance include:

 What it does Type; Precisions; Ranks Syntax Variants; Restrictions Benefits; Bug Warnings Integer powers non-complex x, integer y x ^ y Uses repeated multiplication (avoids log) Powers mod(m) integer, extended integer x m&|@^ y m&|@(n&^) y Avoids the large result of exponentiation

 x ^!.p y Stope Function

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

x ^!.p y is x*(x+p)*(x+2*p)*... for y terms.

```   5 ^!.1 (3)   NB. 5 * 6 * 7
210
```

### Common Uses

1. Calculate the number of permutations of x things taken y at a time, xPy

```   5 ^!._1 (3)   NB. 5 * 4 * 3
60
```

2. Calculate the rising Pochhammer symbol (x)y as used in the hypergeometric function

```   5.1 ^!.1 (3)  NB. 5.1 * 6.1 * 7.1
220.881
```