Find the square root of (-2-3)^2?

  • I
  • Thread starter parshyaa
  • Start date
  • #1
307
19
Root of (-2-3) ^2 = -5 ( because root of squared number is the number itself) but alsoo square of (-2-3) is 25 and its root is (+5) /(-5). Therefore what is the correct answer and reason . I think it is -5(google answer is Also -5) but I don't have any reason. Please help me
 

Answers and Replies

  • #2
35,353
11,678
( because root of squared number is the number itself)
It is not, as your example shows. It is the magnitude of the number.
 
  • #3
Delta2
Homework Helper
Insights Author
Gold Member
3,580
1,387
Both answers 5 and -5 are correct for the square root of (-5)^2.

But to be more accurate when you want the square root of a number , you have to state if you want the negative or the positive square root.

When we just say "square root" we mean by convention the positive square root, so it is "a bit more correct" to say that the (positive) square root of (-5)^2 is 5.
 
  • #4
22,089
3,296
Both answers 5 and -5 are correct for the square root of (-5)^2.

No. This is very wrong. The square root of any number is positive. So the square root of ##(-5)^2## is ##5##.
 
  • #5
Svein
Science Advisor
Insights Author
2,143
694
No. This is very wrong. The square root of any number is positive. So the square root of ##(-5)^2## is ##5##.
As long as you are in the real domain, yes. In the complex domain both +5 and -5 are the square roots of 25 (since there are no "positive numbers").
 
  • Like
Likes parshyaa and Delta2
  • #6
22,089
3,296
As long as you are in the real domain, yes. In the complex domain both +5 and -5 are the square roots of 25 (since there are no "positive numbers").

This is a common definition of the square root in complex numbers, but I don't necessarily agree with it. The problem is that it would make the square root no longer a function, which is undesirable. This is usually fixed by defining a principal square root which only evaluates to ##5## and which has a branch cut (in the same way, our square root in ##\mathbb{R}## is a principal square root too). A nicer solution exists when you go to Riemann surfaces though.
 
  • Like
Likes QuantumQuest and parshyaa
  • #8
307
19
Okk I got it, answer is +/- 5 but we take 5 because of conventional use
 
  • #9
FactChecker
Science Advisor
Gold Member
6,184
2,390
It is standard to use the positive square root of a positive number. In complex analysis, that is called the "principle value" of the square root. The negative value will work but it is not the principle value.

EDIT: If you are doing your own work and taking a square root, you should often consider both the positive and negative values. If both might work, indicate that with ±√. If only the positive should be considered, indicate that with √. If only the negative should be considered, indicate that with -√. In all cases, √ just indicates the positive value.
 
Last edited:
  • Like
Likes jedishrfu, parshyaa and Delta2
  • #10
Svein
Science Advisor
Insights Author
2,143
694
This is a common definition of the square root in complex numbers, but I don't necessarily agree with it. The problem is that it would make the square root no longer a function, which is undesirable. This is usually fixed by defining a principal square root which only evaluates to ##5## and which has a branch cut (in the same way, our square root in ##\mathbb{R}## is a principal square root too). A nicer solution exists when you go to Riemann surfaces though.
But [itex] z^{2}=25\Leftrightarrow z^{2}-25=0 \Leftrightarrow (z+5)\cdot (z-5)=0[/itex]...
 
  • #11
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,361
1,028
Root of (-2-3) ^2 = -5 ( because root of squared number is the number itself) but also the square of (-2-3) is 25 and its root is (+5) /(-5). Therefore what is the correct answer and reason . I think it is -5(google answer is Also -5) but I don't have any reason. Please help me
If you had asked more symbolically, "What is ##\ \sqrt{(-2-3) ^2\,}\, ?\,##" then assuming your context was real rather than complex numbers, the answer would be simply, ##\ \sqrt{(-2-3) ^2\,}=5\ .\ ## In the context of real numbers, the ##\ \sqrt{\ \ } \ ## symbol represents the "principle value" of the square root, as pointed out by FactChecker and others.

Moreover, ##\ \sqrt{x^2\,}=|x| \ ## and not ##\ x\ .\ ## This is often surprising to students. So, the square root of a squared number is not necessarily the number itself.
 
  • #12
pwsnafu
Science Advisor
1,080
85
But [itex] z^{2}=25\Leftrightarrow z^{2}-25=0 \Leftrightarrow (z+5)\cdot (z-5)=0[/itex]...
"Find the square root of 25" and "Find all numbers that square to 25" are different mathematical questions. Solving ##z^{2}=25## is the latter. The former is ##z = \sqrt{25}##.
 
  • Like
Likes Mark44 and parshyaa

Related Threads on Find the square root of (-2-3)^2?

  • Last Post
Replies
3
Views
902
  • Last Post
2
Replies
25
Views
3K
  • Last Post
2
Replies
40
Views
11K
Replies
4
Views
1K
M
  • Last Post
Replies
8
Views
4K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
2
Views
714
Replies
4
Views
2K
Replies
6
Views
31K
  • Last Post
Replies
4
Views
10K
Top