- #1
caters
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Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?
Irrational^rational = rational
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In summary, some irrational numbers raised to rational powers can be rational, such as in the case of √2^4 = 4. However, the irrational number must be algebraic for this to be possible. If the exponent is zero, the result is always rational. But if the exponent is non-zero, then the irrational number must be transcendental and the result will also be transcendental.
- #2
jgens
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Yes. For your first question note that √22 = 2. For your second question all number of this form should be algebraic.
Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.
Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.
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- #3
pwsnafu
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It's also worth pointing out that irrational to the irrational can be rational.
The Gelfond-Schneider theorem says: if ##a## and ##b## are algebraic, with ##a \neq 1,0## and ##b## irrational, then ##a^b## is transcendental.
This means that ##\sqrt{2}^{\sqrt{2}}## is transcendental (hence irrational). Raise this to the power of ##\sqrt{2}## and you get 2.
The Gelfond-Schneider theorem says: if ##a## and ##b## are algebraic, with ##a \neq 1,0## and ##b## irrational, then ##a^b## is transcendental.
This means that ##\sqrt{2}^{\sqrt{2}}## is transcendental (hence irrational). Raise this to the power of ##\sqrt{2}## and you get 2.
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Mark44
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A very simple example is ##\pi^0 = 1##.caters said:
##\pi## is irrational, and 0 is rational.
- #5
rama
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if the order of the root is say x, and the power you raise are multiples of x , then it becomes a rational number(or if the power is 0)
example:√2^4=4
example:√2^4=4
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HallsofIvy
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Note that the irrational number has to be algebraic. A transcendental number raised to a rational power cannot be rational. In fact, it must still be transcendental.
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DrClaude
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HallsofIvy said:
Mark44 said:
Isn't there a contradiction here?
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arildno
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DrClaude said:
I think you just became a candidate for the Fields medal!
- #9
DrClaude
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No , I just became the candidate for another coffee!arildno said:I think you just became a candidate for the Fields medal!
Sorry Mark44, completely missed your humor there...
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Mark44
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That was arildno...DrClaude said:
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DrClaude
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I thought you were trying to be funny by proposing the trivial case "to the power of 0".Mark44 said:
1. What does the term "irrational^rational" mean in mathematics?
The term "irrational^rational" refers to an expression where an irrational number is raised to the power of a rational number. For example, √2^2 would be considered irrational^rational because the square root of 2 is irrational and the power of 2 is rational.
2. Is the result of "irrational^rational" always a rational number?
No, the result of "irrational^rational" is not always a rational number. In fact, it is more likely to be irrational. This is because raising an irrational number to any power (rational or not) will result in an irrational number.
3. Can irrational numbers be raised to irrational powers?
Yes, irrational numbers can be raised to irrational powers. For example, √2^√2 is a valid expression. However, the result will almost always be irrational and cannot be simplified to a rational number.
4. How is "irrational^rational" different from "rational^irrational"?
The difference between "irrational^rational" and "rational^irrational" is the order in which the numbers are raised to each other. In "irrational^rational", the irrational number is the base and the rational number is the exponent. In "rational^irrational", the rational number is the base and the irrational number is the exponent. The result of "rational^irrational" can be rational or irrational, depending on the numbers involved.
5. What are some real-life applications of "irrational^rational"?
"Irrational^rational" can be seen in many real-life situations, such as calculating compound interest or radioactive decay. In these cases, the irrational number represents the growth or decay rate, and the rational number represents the time or number of cycles. It can also be used in physics and engineering for calculations involving power and energy.
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