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11. Power and Inverse

The power conjunction (^:) is analogous to the power function (^). For example:
```   ]a=: 10^ b=: i.5
1 10 100 1000 10000

b
0 1 2 3 4

%:a
1 3.16228 10 31.6228 100

%: %: a
1 1.77828 3.16228 5.62341 10

%: ^: 2 a
1 1.77828 3.16228 5.62341 10

%: ^: 3 a
1 1.33352 1.77828 2.37137 3.16228

%: ^: b a
1      10     100    1000   10000
1 3.16228      10 31.6228     100
1 1.77828 3.16228 5.62341      10
1 1.33352 1.77828 2.37137 3.16228
1 1.15478 1.33352 1.53993 1.77828

(cos=: 2&o.) ^: b d=:1
1 0.540302 0.857553 0.65429 0.79348

] y=: cos ^: _ d
0.739085

y=cos y
1
```
Successive applications of cos appear to be converging to a limiting value; the infinite power (cos ^: _) yields this limit.

A right argument of _1 produces the inverse function. Thus:
```   %: ^: _1 b
0 1 4 9 16

*: b
0 1 4 9 16

%: ^: (-b) b
0 1     2         3         4
0 1     4         9        16
0 1    16        81       256
0 1   256      6561     65536
0 1 65536 4.30467e7 4.29497e9
```

Exercises

 11.1 The square function *: is the inverse of the square root function %: and %:^:_1 is therefore equivalent to *: . Look for, and experiment with, other inverse pairs among the primitive functions in the Vocabulary.

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