Solving -x^n: Pre-Calculus Confusion

[Husaaved]

Hello,

This is my first post here. I am sorry if this is the wrong place for this question or if my phrasing is unclear.

-x^n≠(-x)^n
-x^n=-(x^2)

I am slightly confused by this because at the level I am at right now (i.e. pre-calculus) and before, I have seen statements that contradict this. For example, I feel like I have been told from beginning algebra until now that -2^2=4. When is this true and when is it not?

I suppose it is a matter of notation?

Thank you.

 

[FeDeX_LaTeX]

Depends on how you place the parentheses.

$(-2)^2 = (-2) \cdot (-2) = 4$
$-(2^2) = -[(2) \cdot (2)] = -4$

In general, $-(x^n) = (-x)^n$ if n is an odd integer.

 

[Husaaved]

Thank you for your swift response.

If there are no parentheses, what is assumed? Or does that vary from text to text?

 

[FeDeX_LaTeX]

I'd say that if there are no parentheses given, it's safe to assume that they mean $-x^n = -(x^n)$, unless they explicitly say otherwise.

 

[Mark44]

-x^n≠(-x)^n
-x^n=-(x^2)

Probably a typo in the 2nd equation. I guess you mean -x^n = -(x^n).

I am slightly confused by this because at the level I am at right now (i.e. pre-calculus) and before, I have seen statements that contradict this. For example, I feel like I have been told from beginning algebra until now that -2^2=4. When is this true and when is it not?

I suppose it is a matter of notation?

You should not have been told that -2² = 4. You might be misremembering what was said in class.

I'd say that if there are no parentheses given, it's safe to assume that they mean $-x^n = -(x^n)$, unless they explicitly say otherwise.

I edited my earlier response. What you're saying is correct, FedEx_LaTeX, but you don't need the qualifier. $-x^n$ and $-(x^n)$ mean exactly the same thing.

According to the operator precedence, exponents are evaluated before signs.
-xⁿ means the opposite of xⁿ, or -(xⁿ). If you want to raise -x to a power you have to include parentheses around the thing being raised to the power, like this: (-x)ⁿ.

 

[baker0]

In addition, be careful about the first statement that you made.
-xⁿ≠(-x)ⁿ does not hold for all n and x that you are used to using.
This statement is FALSE for all odd n. There is also some x for which this is false.

 

[economicsnerd]

No, that is incorrect. According to the operator precedence, exponents are evaluated before signs.
-xⁿ means the opposite of xⁿ, or -(xⁿ). If you want to raise -x to a power you have to include parentheses around the thing being raised to the power, like this: (-x)ⁿ.

Mark, I believe what you've written agrees with what FeDeX wrote.

 

[Mark44]

Mark, I believe what you've written agrees with what FeDeX wrote.

I now see that, and have revised what I wrote in that thread. My only excuse is that the "unless they explicitly say otherwise" part threw me off.

I'm so used to seeing people write, for example, -2² = +4, that I thought (erroneously, mea culpa) that that's what FedEx_LaTeX was saying. Hopefully, things are clear now.