Why is sqr rt (16x^2) both positive and negative?

Because squaring of both leads to same answer

√(16x²) = ±4x

Now check !
(+4x)² = 16x²
(-4x)² = 16x²

The positive sign in front of a positive number can be omitted, e.g. 3 = +3.  Likewise. √(16x²) means +√(16x²).  There is NO negative value for √(16x²).

Hence, √(16x²)
= √(4x)²
= 4x

Confusion is made because of the following case:
y² = 16x²
y = ±√(16x²)
y = ±4x

Do you notice the difference?

Do you mean √(16x²)? That symbol is usually called the square root, but strictly speaking it indicates the principal square root. It can be positive, or it can be zero, but it cannot be negative.

√(16x²) = 4|x|

It is true that a negative number can be a square root. However, it cannot be the principal square root of any number.

In answer to your question, why is it both positive and negative? It is not.

Assuming "sqr rt" means "square root":

If x is a real number, then x^2 is positive or zero, so 16x^2 is positive or zero.

The "square root" of zero is zero.

Usually, the "square root" of a positive number means the "principal" or "positive" square root unless otherwise specified. So if 16x^2 is positive, then sqrt(16x^2) is positive.

That's different from saying y^2 = 16x^2 and you want to find y; in that case, there are two solutions for y (unless x = 0).

Note that if x is a non-real number, then 16x^2 is not positive or negative.

The idea [existence] of a negative sqrt answer arises because a negative times a negative is a positive. The idea is also supported by geometry, where going left on the X axis is considered a negative number, and going down / south on the Y axis is negative; but when you multiply you get a positive amount of area.  "No negative sqrt" is a new math convention/"rule" because computers cannot handle 2 answers to the same problem, like an old school algebra person can.  In computer language (i e PEDMAS) there are NO radical signs. You would say (16x^2)^(1/2). {of course, the x squared has to be a real number ... to be multiplied by 16, and then the sqrt is computed.}