Lesson: Introduction to Square Roots

That's correct.

Question link: https://gre.myprepclub.com/forum/topic11876.html

You are correct. The square root notation (√), tells us to take POSITIVE value only.
For example, √25 = 5, √81 = 9 and √0.25 = 0.5

IMPORTANT: Some people conflate the above notation issue with equations involving powers of 2.

For example, the equation x² = 9 does not have any square root notation.
So, we must recognize that there are TWO possible solutions: x = 3 and x = -3.

Likewise, the equation x² = 100 also has TWO possible solutions: x = 10 and x = -10.

In general, we can write: If x² = k, then there are two solutions: x = √k and x = -√k

Does that help?

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/topic11876.html

Your calculations are missing a zero.
(0.06)² = 0.0036 (not 0.036)

Cheers,
Brent

Question link: https://gre.myprepclub.com/forum/x-is-an-integer-11296.html

The problem is that testing numbers will never yield definitive results UNLESS the correct answer is D.
I cover this at 2:50 in the following video: https://www.greenlighttestprep.com/module/gre-quantitative-comparison/vi...

Having said that, if you keep testing numbers and you get the same result each time (e.g., Quantity A is greater), then that certainly SUGGESTS that the correct answer is A.

Does that help?

That's a great (clever) question.
Regardless of how the expression is set up, there are still implied some orders of operation.
Here's what I mean:

The answer to that question depends on whether there are any additional brackets. Here are 2 cases:

case i: [sqrt of (-25)]^2
In other words, [√(-25)]²
In this case, we must evaluate √(-25) first
Since there is no (real) square root of -25, we can conclude that [√(-25)]² has no real value.

case ii: sqrt of [(-25)]^2
In other words, √[(-25)²]
In this case, we must evaluate (-25)² first
(-25)² = 625, so √[(-25)²] = √625 = 25

ASIDE: I should also mention that in your solution, you are taking the square root TWICE.
You have: sqrt of (-25)^2 = [(-5)^2]^1/2, when it should be sqrt of (-25)^2 = [(-25)^2]^1/2

Does that help?

General property #1:
If √(x²) = k, where k > 0, then x = k or x = -k

General property #2:
If (√x)² = k, where k > 0, then x = k

Q: Why does general property #2 yield only one value for x?
A: Because we can't take the square root of a negative value.

The square root of -5 does not have a real value.
If you're referring to 5:25 in the lesson, keep in mind that we are finding the square root of (-5)² and not the square root of -5.
Since (-5)² = 25, we are really taking the square root of 25.

Does that help?