What is the value of $\sqrt{(-4)^2}$? (Sorry for the triviality!)

Question

I started a debate in a youtube video (something you should never do...) because of this question. For me, the answer has to be $-4$, simplifying powers for example. But I don't know if I'm having serious definition failures. Another possibility would be to swap the powers of order and do the root first, taking the complex value and squaring it. However for the vast majority of people who commented, the result was $+4$.

Sorry for the triviality and thanks for being the best mathematics forum in the world.

Answer 1

I don't feel that the linked answers completely solves you problem, so let me spell it out a little bit.

You write in your comments that

$\begin{equation} \sqrt{x^2}=(x^2)^{\frac{1}{2}}=x \end{equation}$

This is not correct. Sure, for a real number $x$ and integers $a$ and $b$, you have $(x^a)^b=x^{ab}$, but this is not true for non-integer exponents.

What might have confused you, is that the above rule is true when $x$ is a positive real number, and a lot of texts might ommit this crucial assumption when doing symbolic manipulations as the one suggested in the wrong equation above.

As some comments suggest, the correct equation is

$\sqrt{x^2}=|x|$

You can maybe convince yourself why this is true by considering what happens when $x$ is either negative or positive. Else let me know, and I'll happily elaborate.

Edit: Comprehensive proof that $\sqrt{x^2}=|x|$.

Let $x$ be any real number. If $x\ge 0$, we have by definition of the squareroot function that

$\sqrt{x^2}=x=|x|$

The first equality is true, since $x$ is the unique non-negative number satisfying the equation $y^2=x^2$ (where we solve for $y$). The second equality is true since the absolute value of a non-negative number is the number itself.

Let's now assume that $x< 0$. By this assumption, we have that $|x|=-x$. From this, we get

$x^2=1\cdot x^2=(-1)^2x^2=(-x)^2=|x|^2$

From this we get that

$\sqrt{x^2}=\sqrt{|x|^2}$

Since $|x|$ is a positive number, we conclude from the previous that

$\sqrt{x^2}=\sqrt{|x|^2}=|x|$

Answer 2

There is a flaw in your argument,, Because if your argument is true then 4 =√16=√((-4)^2)= -4 NOW how can 4=-4 ,, there is a flaw. So you have to choose either √16 = 4 or -4, So let's do another experiment If √(16) = -4 Then let's calculate √(16)×√(16) = -4×-4=16, but it can also be written as √(16×16) =√(-16)^2= -16 How come 16=-16 ,, Look I can give you tonnes of examples like this ,your logic is correct but if we were to use your rule then whole mathematics will collapse just like this . Any rule you make should not only be logical but also consistent with previous rules.

So in conclusion your logic is correct but it just doesn't fit in right with other rules of mathematics..

Answer 3

For the notation $\sqrt{}$ to make any sense, it must be a well-defined function - that is, for any input $x$, it must give one, and only one result for $\sqrt{x}$.

I think we can all agree that $\sqrt{16}=+4$. We have to pick either $+4$ or $-4$, because remember, we need one and only one answer, so let's arbitrarily pick $+4$ (following normal convention).

However, I think we can all recognize the fact that $(-4)^2=16$. Therefore, $\sqrt{(-4)^2}=\sqrt{16}=+4$. It cannot be true that $\sqrt{(-4)^2}=-4$, because if that was the case, it would contradict our previously established fact that $\sqrt{16}=+4$, since $(-4)^2=16$.

In summary, the claim that $\sqrt{(-4)^2}=-4$ while simultaneously $\sqrt{16}=4$ violates the rules of logic.