Can "Real Powers" Give You a Rational Number?

Thread starter Derivative86
Start date Jan 9, 2004
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Rational

In summary, real powers, or exponential functions, can produce both rational and irrational numbers. A rational number is any number that can be expressed as a ratio of two integers, while an irrational number cannot be expressed as a ratio and has an infinite number of non-repeating decimals. Real powers can only generate rational numbers when the exponent is a rational number itself, meaning that there is a limit to the rational numbers that can be produced by real powers. However, real powers can still produce irrational numbers when the exponent is an irrational number. Real powers and rational numbers are closely related through the concept of exponents, as real powers involve raising a base number to a power and can generate rational numbers when the exponent is rational.

Jan 9, 2004

Derivative86

can e to any real powers give you a rational number?

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Jan 9, 2004

NateTG

Science Advisor

Homework Helper

You mean like?
[tex]e^{\ln x} : x \in \mathbb{Q}[/tex]

Nah, that couldn't be it.

Jan 9, 2004

Derivative86

Actually, u r right... i was being so stupid lol

1. Can real powers give me any rational number?

Yes, real powers can give you rational numbers. A rational number is any number that can be expressed as a ratio of two integers, such as 3/4 or 5/2. Real powers, or exponential functions, can produce rational numbers when the exponent is a rational number itself.

2. Is there a limit to the rational numbers that can be generated by real powers?

Yes, there is a limit. Real powers can only produce rational numbers when the exponent is a rational number. This means that there are an infinite number of rational numbers that can be generated by real powers, but they are limited to specific values.

3. How do I know if a number is rational or irrational?

A rational number can be expressed as a ratio of two integers, while an irrational number cannot be expressed as a ratio and has an infinite number of non-repeating decimals. To determine if a number is rational or irrational, you can try to express it as a fraction. If it can be expressed as a fraction, then it is rational. If not, then it is irrational.

4. Can real powers ever produce an irrational number?

Yes, real powers can produce irrational numbers. This happens when the exponent is an irrational number, such as pi or the square root of 2. When an irrational number is raised to a power, the result will also be irrational.

5. How are real powers and rational numbers related?

Real powers and rational numbers are related through the concept of exponents. Real powers, or exponential functions, involve raising a base number to a power, which can be a rational or irrational number. Rational numbers, on the other hand, can be generated by real powers when the exponent is a rational number. In summary, real powers can produce rational numbers, but not all rational numbers can be generated by real powers.