A very quick stupid question about square roots of squares

In summary, the square root of an expression squared does not necessarily equal the expression itself. This is because the square root function always returns a positive value, and an expression squared can lose its signed identity. Therefore, you cannot simply remove the square root sign and replace it with the expression.
  • #1
help1please
167
0


If I see an expression like

[tex]\sqrt{E^2c^2}[/tex]

I can just remove the square root sign right and replace it with Ec?
 
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  • #2


In general, no.

One must recall that the square root function always returns a positive value. So,
[tex]\sqrt{X^{2}} = |X|[/tex]
This is aptly illustrated by [itex]\sqrt{(-1)^{2}} = |-1| = 1[/itex].
[itex]\sqrt{X^{2}} = X[/itex] iff X is positive.

(for simplicity, assume that X is always real)
 
  • #3


help1please said:
If I see an expression like

[tex]\sqrt{E^2c^2}[/tex]

I can just remove the square root sign right and replace it with Ec?

You can't. And please improve the title a bit, like the one you see in the title of this post.
 
  • #4


In general, you cannot do that. If however, if you know for certain that E and c are both real and positive numbers, then yes you can simplify it to just Ec.
 
  • #5


help1please,

I can just remove the square root sign right and replace it with Ec?

I would say yes you can. Once a expression is squared, it loses its signed identity.

Ratch
 
  • #6


Ratch said:
help1please,

I would say yes you can. Once a expression is squared, it loses its signed identity.
That is incorrect. Here's an example why.

$$ \sqrt{(-4)^2} \neq -4$$

IOW, if you square a negative number, and then take the square root, you don't get the negative number back.
 
  • #7


Mark44,

IOW, if you square a negative number, and then take the square root, you don't get the negative number back.

Correct, that is what I am averring. I am saying it loses its sign identity after it is squared.

Ratch
 
  • #8


Ratch said:
Mark44,
Correct, that is what I am averring. I am saying it loses its sign identity after it is squared.
That's not what you told the OP.

help1please said:
I can just remove the square root sign right and replace it with Ec?

Ratch said:
I would say yes you can. Once a expression is squared, it loses its signed identity.

So what you are telling the OP is that
$$ \sqrt{E^2c^2} = Ec$$

And that is true only if both E and c are nonnegative. If either is negative, the above is not true.
 
  • #9


Mark44,

Re: A very quick stupid question

--------------------------------------------------------------------------------

Originally Posted by Ratch
Mark44,
Correct, that is what I am averring. I am saying it loses its sign identity after it is squared.

That's not what you told the OP.

That is exactly what I told the OP. See post #5.

So what you are telling the OP is that

No, I am saying that √(E^2*C^2) = |E*C|

Ratch
 
  • #10


Ratch said:
Mark44,
That is exactly what I told the OP. See post #5.
Yes, I saw post #5, and that is what I'm objecting to.

The OP's question:

help1please said:
I can just remove the square root sign right and replace it with Ec?
IOW, help1please is asking if ##\sqrt{E^2c^2} = Ec##

Your verbatim response in post #5 (again):
Ratch said:
I would say yes you can. Once a expression is squared, it loses its signed identity.
Ratch said:
No, I am saying that √(E^2*C^2) = |E*C|
Well, I agree with that, but that isn't what you said before. Your previous response had no mention of absolute values.
 
  • #11


Mark44,

OK, you are right. I only half explained it.

Ratch
 

Related to A very quick stupid question about square roots of squares

1. What is a square root?

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 multiplied by itself equals 9.

2. What is the difference between the square root and the square of a number?

The square root of a number is the number that, when squared, gives the original number. The square of a number is the result of multiplying the number by itself. For example, the square root of 9 is 3, and the square of 3 is 9.

3. Can you take the square root of a negative number?

No, the square root of a negative number is not a real number. The square root of a negative number is called an imaginary number, denoted by the letter "i".

4. What is the difference between the principal square root and the negative square root?

The principal square root is the positive square root of a number. For example, the principal square root of 9 is 3. The negative square root is the negative version of the principal square root. So, the negative square root of 9 is -3.

5. How can you simplify square roots?

To simplify a square root, you can factor the number into smaller numbers and take the square root of each individual factor. For example, to simplify the square root of 12, you can factor it into 4 and 3, and then take the square root of each factor separately, giving you √4 * √3, which simplifies to 2√3.

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