Is x^{\sqrt{5}}=y for Rational Numbers?

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In summary, the expression "Rational ^ sqrt(5) = rational" means that applying the power of the square root of 5 to a rational number will result in another rational number. This is different from the standard representation of a rational number as a fraction. An example of a rational number raised to the square root of 5 is (4/9)^sqrt(5) = (4/9)^(2.236) = 0.481. However, this expression is not always true and its significance lies in its applications in various fields of mathematics and real-world problems.
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Gerenuk
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I've programmed an algorithm to numerically compute the logarithm of numbers in phinary base easily. I could avoid float multiplications if I can find a pair of rational numbers x and y such that
[tex]x^{\sqrt{5}}=y[/tex]
Is it possible?
Probably not, but I cannot prove it :(
 
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No, it isn't possible except for the obvious x=y=1. There's a http://en.wikipedia.org/wiki/Gelfond–Schneider_theorem" which says that it's impossible.
 
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FAQ: Is x^{\sqrt{5}}=y for Rational Numbers?

What does the expression "Rational ^ sqrt(5) = rational" mean?

The expression means that the power of the square root of 5 when applied to a rational number will result in another rational number.

How is this expression different from the standard rational number representation?

The expression is different because it involves using a power function on a rational number, rather than the traditional representation of a fraction.

Can you provide an example of a rational number raised to the square root of 5?

Yes, an example would be (4/9)^sqrt(5) = (4/9)^(2.236) = 0.481.

Is this expression always true for any rational number?

No, this expression is not always true. It depends on the value of the rational number and the power of the square root of 5 being applied to it.

What is the significance of this expression in mathematics or real-world applications?

This expression has applications in various fields such as geometry, engineering, and physics. It can be used to solve problems involving geometric shapes and calculations, as well as in the analysis of complex systems and data.

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