Square Root vs Cube Root

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Discussion Overview

The discussion revolves around the differences between square roots and cube roots, particularly focusing on why square roots yield both positive and negative answers while cube roots yield only a positive answer. Participants also explore related algebraic expressions and the implications of complex numbers in root calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant notes that x^2 = 4 yields two solutions (x = -2 or x = 2), while x^3 = 8 yields only x = 2, prompting a question about the underlying reasons for this difference.
  • Another participant explains that the product of two positive or two negative numbers is positive, while the product of three negative numbers is negative, suggesting a relationship to the nature of roots.
  • Discussion on complex numbers indicates that every number has n nth roots, with the nature of these roots varying based on whether n is even or odd, affecting the number of real roots.
  • Several participants discuss the expression sqrt{x^6}, debating whether the answer should be x^3 or -x^3, with some suggesting that it should be expressed as |x^3| for clarity.
  • One participant asserts that if x is non-negative, sqrt{x^6} is x^3, while if x is negative, it is -x^3, reinforcing the idea that sqrt{x^6} is |x^3|.
  • Another participant questions why |x^3| is the correct expression, seeking further clarification on the reasoning behind this notation.

Areas of Agreement / Disagreement

Participants express differing views on how to handle the expression sqrt{x^6}, with some advocating for |x^3| while others emphasize the conditions under which x^3 or -x^3 should be used. The discussion remains unresolved regarding the best way to express the results of square roots in algebraic terms.

Contextual Notes

There are limitations in the discussion regarding assumptions about the values of x, particularly in relation to non-negative and negative cases, as well as the implications of complex roots which are not fully explored.

mathdad
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I know that x^2 = 4 yields two answers: x = -2 or x = 2.

I also know that x^3 = 8 yields x = 2.

Question:

Why does the square root yield both a positive and negative answer whereas the cube root yields a positive answer?
 
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The product of two positive or two negative numbers is positive. The product of three positive numbers is positive but the product of three negative numbers is negative.

By the way, if we allow complex numbers, then every number has n nth roots. If the original number is a positive real number, those roots lie on a polygon in the complex plane with n vertices, one of them the positive real root. If n is even, then there is another root on the negative real axis (for example, if n= 4 that is a square with one diagonal being the real axis) If n is odd the only real root is positive (if n= 3 we have an equilateral triangle and the real axis goes through the middle of the side between the two non-real roots.

Also, in an equation with all real coefficients, for every non-real root, its complex conjugate is also a root. That means there is always an even number of non-real roots. Since the principal root is real, if n is odd that is the only real root, if n is even, there is another real root.
 
What about algebraic terms?

Say, sqrt{x^6}.

Can we say the answer is x^3 or -x^3 and x^3?
 
RTCNTC said:
What about algebraic terms?

Say, sqrt{x^6}.

Can we say the answer is x^3 or -x^3 and x^3?

If $0\le x$, then we can call it $x^3$, otherwise, we call it $\left|x^3\right|$. :D
 
That strikes me as an odd way of phrasing it. If x\ge 0 then \sqrt{x^6} is x^3. If x< 0 then \sqrt{x^6}is -x^3. In either case, \sqrt{x^6} is |x^3|.
 
MarkFL said:
If $0\le x$, then we can call it $x^3$, otherwise, we call it $\left|x^3\right|$. :D

If 0 is < or = x, then the answer is x^3. Otherwise, the answer must be the absolute value of x^3. Why is |x^3| the correct way to express the answer?
 
RTCNTC said:
If 0 is < or = x, then the answer is x^3. Otherwise, the answer must be the absolute value of x^3. Why is |x^3| the correct way to express the answer?

Here is an article on the properties of square roots:

Square root
 
Interesting.
 

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