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 Powers of powers (Posted on 2004-05-11)
Solve for x, if:
```   3
x
x
x    = 3
```
An algebraic solution is sought!

 See The Solution Submitted by Federico Kereki Rating: 4.1000 (10 votes)

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 Power Tower | Comment 5 of 11 |

Substitute x^(x^(x^3)) for the three in the exponent to get:
x^(x^(x^(x^(x^(x^3))))) = 3
x^(x^(x^(x^(x^(x^(x^(x^(x^3)))))))) = 3

And repeat ad infinitum to get an infinite power tower:
x^(x^(x^(x^(x^(x^(x^(x^(x^(.........))))))))) = 3

Now the exponent of the bottom x is identical to the entire expression, so 3 can be substituted to yield x^3 = 3

The value x = cbrt(3) is the solution to all of the equations.

 Posted by Brian Smith on 2004-05-11 15:29:34
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