Rational exponents in the real number system?

In summary, the conversation discusses the topic of raising real numbers to rational powers without using complex numbers. The goal is to find authoritative answers and definitions for this topic in a self-consistent manner, without relying on complex analysis. The conversation also mentions a lack of rigorous treatment of this topic in current texts, with an example of "baby Rudin" not providing a definition for exponentiation of negative numbers. The speaker suggests a text on real analysis that would provide a definition for this topic.
  • #1
Stephen Tashi
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Are there rigorous texts that treat the topic of raising real numbers to rational powers without treating it a special case of using complex numbers?

I'm not trying to avoid the complex numbers for my own personal use! My goal is to determine whether students who have not studied complex analysis can find authoritative answers.

In casual discussions, people are quick to offer answers to questions such as:
1) Is ##(-8)^{4/6} ## equal to ##(-8)^{2/3}##?
2) Is ##(1)^{5/0}## defined as the 5th power of the zeroth root of 1?

but what texts give authoritative answers to such questions and develop definitions for the above notations in a manner that is self-consistent?

Usually when the topic of raising a negative number to a rational power comes up, the answers refer to complex numbers. Has any author taken up the thankless task of treating this topic in a rigorous fashion without using complex analysis?
 
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  • #2
Stephen Tashi said:
in a rigorous fashion without using complex analysis?

"Rigor" means different things to different people. What do you think is legit and what do you not? In short - can you describe what an answer would look like?
 
  • #3
Vanadium 50 said:
"Rigor" means different things to different people. What do you think is legit and what do you not? In short - can you describe what an answer would look like?

I imagine a text on "Real analysis" that defines the exponentiation of negative numbers. As a non- example, I don't find such a definition in "baby Rudin" ( Principles of Mathematical Analysis ).
 

FAQ: Rational exponents in the real number system?

What are rational exponents?

Rational exponents are exponents that are expressed as a fraction, where the numerator is the power and the denominator is the root. For example, the expression 21/2 is a rational exponent, where the numerator 1 is the power and the denominator 2 is the square root.

How do you simplify expressions with rational exponents?

To simplify expressions with rational exponents, you can use the properties of exponents. For example, the expression 82/3 can be simplified to (23)2/3 = 26/3 = 22 = 4.

Can rational exponents be negative?

Yes, rational exponents can be negative. For example, the expression 4-1/2 is a rational exponent, where the numerator -1 is the power and the denominator 2 is the square root. This can be rewritten as 1/41/2, which is equal to 1/√4 = 1/2.

How do you convert rational exponents to radical form?

To convert rational exponents to radical form, you can use the property am/n = n√am. For example, the expression 272/3 can be written as 3√272 = 3√(272) = 3√(729) = 9.

Can rational exponents be simplified to whole numbers?

Yes, rational exponents can be simplified to whole numbers. For example, the expression 83/3 can be simplified to (23)3/3 = 29/3 = 23 = 8.

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