What is the square root of x^2?

In summary, the conversation discusses the concept of taking the square root of a number, and whether it should always return a positive value or if it can have both positive and negative values. It is agreed that when referring to "the square root" of a number, it is usually referring to the principal or positive value. However, when solving equations, both positive and negative values should be considered. It is also noted that the square root function is defined as a single-valued function, and taking the negative square root does not provide new information.
  • #36
Benn said:
I think you might mean

[tex]{ x }^{ 2 }=4\\ \sqrt{{ x }^{ 2 }}=\sqrt { 4 } \\ |x| = 2 \\ x=\pm 2[/tex]

As a reminder for the OP, while this argument does prove
If x2 = 4, then x = 2 or x = -2​
it does not prove
x = 2 and x = -2 are both solutions to x2 = 4​
unless you can argue that every step is reversible. (they are in this case)
 
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  • #37
Hurkyl said:
As a reminder for the OP, while this argument does prove
If x2 = 4, then x = 2 or x = -2​
it does not prove
x = 2 and x = -2 are both solutions to x2 = 4​
unless you can argue that every step is reversible. (they are in this case)

What do you mean by every step is reversible? Can you give me an example where they are not reversible?
 
  • #38
x = -2 ## \Rightarrow## x2 = 4

The steps here are not reversible. If x2 = 4, it does not necessarily imply that x = -2.
 
  • #39
Hurkyl said:
As a reminder for the OP, while this argument does prove
If x2 = 4, then x = 2 or x = -2​
it does not prove
x = 2 and x = -2 are both solutions to x2 = 4​
unless you can argue that every step is reversible. (they are in this case)

Wrong.Is x=2 a solution to x^2=4?-yes and is x=-2 a solution to x^2=4-yes, they are both solutions.Saying x=two values simultaneously is wrong, which I think is what you meant and thinking about this, with everything said generally being correct in this thread about(x^0.5)^2 being= to + or-(x) or abs(x) this means that (x^a)^b doesn't always equal x^(ab), which is an exponentiation law-I suppose there always are exceptions.The steps are irreversible because the inverse has more than one image, which isn't right.
 
  • #40
Dalek1099 said:
Wrong.Is x=2 a solution to x^2=4?-yes and is x=-2 a solution to x^2=4-yes, they are both solutions.Saying x=two values simultaneously is wrong, which I think is what you meant and thinking about this, with everything said generally being correct in this thread about(x^0.5)^2 being= to + or-(x) or abs(x) this means that (x^a)^b doesn't always equal x^(ab), which is an exponentiation law-I suppose there always are exceptions.The steps are irreversible because the inverse has more than one image, which isn't right.

What Hurkyl said was absolutely correct. I think you misinterpreted his post.
 
  • #41
Alright, thanks for the help guys.
 
<h2>1. What is the square root of x^2?</h2><p>The square root of x^2 is x. This is because the square root of a number is a value that, when multiplied by itself, gives the original number. In this case, x multiplied by itself (x^2) gives x^2, therefore the square root of x^2 is x.</p><h2>2. Can the square root of x^2 be negative?</h2><p>No, the square root of a number cannot be negative. This is because a negative number multiplied by itself will always result in a positive number. Therefore, the square root of x^2 will always be a positive value, regardless of the value of x.</p><h2>3. How can I calculate the square root of x^2?</h2><p>The square root of x^2 can be calculated by taking the square root of x and then squaring the result. In other words, the square root of x^2 is equal to the absolute value of x. For example, if x = -4, the square root of x^2 would be |-4| = 4.</p><h2>4. Is the square root of x^2 the same as x?</h2><p>Yes, the square root of x^2 is the same as x. This is because the square root of a number is the value that, when multiplied by itself, gives the original number. In this case, x multiplied by itself (x^2) gives x^2, therefore the square root of x^2 is x.</p><h2>5. Can the square root of x^2 be a fraction or decimal?</h2><p>Yes, the square root of x^2 can be a fraction or decimal. This is because the square root of a number can be any value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, but the square root of 9.61 is 3.1. However, it should be noted that the square root of x^2 will always be a positive value, regardless of whether it is a fraction or decimal.</p>

1. What is the square root of x^2?

The square root of x^2 is x. This is because the square root of a number is a value that, when multiplied by itself, gives the original number. In this case, x multiplied by itself (x^2) gives x^2, therefore the square root of x^2 is x.

2. Can the square root of x^2 be negative?

No, the square root of a number cannot be negative. This is because a negative number multiplied by itself will always result in a positive number. Therefore, the square root of x^2 will always be a positive value, regardless of the value of x.

3. How can I calculate the square root of x^2?

The square root of x^2 can be calculated by taking the square root of x and then squaring the result. In other words, the square root of x^2 is equal to the absolute value of x. For example, if x = -4, the square root of x^2 would be |-4| = 4.

4. Is the square root of x^2 the same as x?

Yes, the square root of x^2 is the same as x. This is because the square root of a number is the value that, when multiplied by itself, gives the original number. In this case, x multiplied by itself (x^2) gives x^2, therefore the square root of x^2 is x.

5. Can the square root of x^2 be a fraction or decimal?

Yes, the square root of x^2 can be a fraction or decimal. This is because the square root of a number can be any value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, but the square root of 9.61 is 3.1. However, it should be noted that the square root of x^2 will always be a positive value, regardless of whether it is a fraction or decimal.

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