Solving a Basic Math Contradiction

In summary, the conversation discusses a contradiction that arises when computing the square root of a negative number raised to different powers. The "laws of exponents" do not apply to complex numbers in the same way as to real numbers. One reply in a previous thread suggests that the numbers being raised must be mutually prime, but this requirement is not explicitly stated and may be due to avoiding contradictions. The conversation also touches on the issue of using equalities when computing with complex numbers.
  • #1
Aeneas
27
0
Could someone please sort out this contradiction which must come from some very basic error - but where and which error? If you raise -3 to the power of 1/2, this gives the square root of -3 which has no real value, but if you raise it to the power of 2/4, you are finding the fourth root of -3 squared, which is the fourth root of +9 which is real. What is wrong here?

Thanks in anticipation.
 
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  • #3
Roughly speaking, the "laws of exponents" do not apply to complex numbers in the same way they apply to real numbers. But certainly look at the link Diffy mentioned.
 
  • #4
Thanks for that, but I do no fully follow the replies in the link. One reply says that when you write p[tex]^{a/b}[/tex], a and b must be mutually prime. The demonstration that p[tex]^{a/b}[/tex]= [tex]\sqrt{p^{a}}[/tex]seems to work whether they are or not.

e.g. p[tex]^{a_{1}/b}[/tex] X p[tex]^{a_{2}/b}[/tex] ...X p[tex]^{a_{b}/b}[/tex] = p[tex]^{ab/b}= p^{a}[/tex].

Thus p[tex]^{a/b}[/tex] = [tex]\sqrt{p^{a}}[/tex]. Where is the requirement there that they should be mutually prime? Or is it that the requirement is created by the need not to get into the contradiction?
 
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  • #5
Aeneas said:
Thanks for that, but I do no fully follow the replies in the link. One reply says that when you write p[tex]^{a/b}[/tex], a and b must be mutually prime.

I am not sure where that person was going with that reply. Certainly one can compute an answer for [tex]64^{2/6}[/tex].
I think that what he was getting at is that you are in tricky waters when you start using equalities. For example [tex]64^{2/6}[/tex] and [tex]64^{1/3}[/tex] aren't necessarily equal. Consider the polynomials that these two expressions are solutions to, Sqrt(x^6) = 64 and x^3 = 64. For the first wouldn't you say the answer is 4 or -4? and for the second there is only one answer 4. Therefore how could you say the two statements are equal?
 

Related to Solving a Basic Math Contradiction

What is a basic math contradiction?

A basic math contradiction occurs when two statements or equations have opposite or conflicting results, making it impossible for both statements to be true at the same time.

Why is it important to solve a basic math contradiction?

Solving a basic math contradiction is important because it allows us to identify and correct errors in our mathematical reasoning, ensuring that our calculations and solutions are accurate and reliable.

How do you solve a basic math contradiction?

The first step to solving a basic math contradiction is to carefully examine the two conflicting statements or equations and identify any errors or inconsistencies. Then, use mathematical operations and properties to manipulate the equations and reach a new statement that is logically consistent and resolves the contradiction.

What are some common mistakes when solving a basic math contradiction?

Some common mistakes when solving a basic math contradiction include overlooking or misinterpreting key information, using incorrect mathematical operations or properties, and making calculation errors.

Are there any tips for avoiding basic math contradictions?

To avoid basic math contradictions, it is important to double-check all calculations and equations, carefully analyze and interpret information, and seek help or clarification when needed. It is also helpful to practice and become familiar with common mathematical operations and properties.

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