Square roots of positive numbers

Click For Summary

Homework Help Overview

The discussion revolves around the properties of square roots, particularly in the context of positive real numbers and the implications of the equation λ² = ab, where a and b are positive. The original poster expresses confusion regarding the presence of negative values in the square root of a product of two positive numbers.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of square roots and the implications of squaring negative numbers. They discuss the distinction between the square root itself and the potential for negative solutions when considering the equation λ² = ab.

Discussion Status

The conversation has led to some clarifications regarding the concept of square roots and the reasoning behind the presence of negative solutions. Participants have provided examples and explanations that seem to aid in understanding the original poster's confusion.

Contextual Notes

The original poster's question reflects a common misunderstanding about square roots and their properties, particularly in relation to positive numbers and the implications of squaring negative values.

bluskies
Messages
11
Reaction score
0

Homework Statement



If a and b are positive real numbers, and \lambda^{2} = ab, then \lambda = \pm \sqrt{ab}.

Homework Equations



None.

The Attempt at a Solution



This is more of a conceptual question that has always escaped me. I do not understand how the square root of two positive numbers could possibly be negative. Since a and b are positive, how can there be any negatives in the square root of their product? Any guidance on this subject would be very much appreciated.
 
Physics news on Phys.org
(-1)2 =1

so squaring -√(ab) will give (ab) regardless of a + or - before the square root.
 
I think you might be looking too hard into this. There are no negatives in the square root. the negative is outside it. say both a and b are 5. then lambda² = 25. lambda therefore is equal to positive root(25) or negative root(25). Or in other terms, lambda² = 5² or (-5)²
 
A numerical example can illustrate this. Let a = 1 and b = 4 for instance. Then we have \lambda^{2} = 4. We realize however, that because negatives cancel upon multiplication, in fact (-2)^2 = 2^(2) = 4, and so both -2 and 2 are possible solutions.
 
Thank y'all for your help, that makes sense now. :)
 

Similar threads

Adding square roots of ##i## leading to different answers!
  • · Replies 21 ·
Replies
21
Views
2K
An inequality involving ##x## on both sides: ##\sqrt{x+2}\ge x##
  • · Replies 3 ·
Replies
3
Views
2K
Show that square root of 3 is an irrational number
  • · Replies 1 ·
Replies
1
Views
2K
Quadratic equation with no rational roots
  • · Replies 12 ·
Replies
12
Views
3K
To prove that a given quadratic has integral roots
  • · Replies 20 ·
Replies
20
Views
3K
Is a^2 Always Positive for a Real Number a?
  • · Replies 18 ·
Replies
18
Views
3K
Proving that there is no rational number whose square is two
  • · Replies 3 ·
Replies
3
Views
2K
Taking negative/positive square roots
  • · Replies 4 ·
Replies
4
Views
2K
When is the root of a number both negative and positive?
  • · Replies 4 ·
Replies
4
Views
2K
Find the values of b for which the roots are positive
  • · Replies 11 ·
Replies
11
Views
2K