Kancz44 wrote:SqRt of X^2 should equal |X| which is equal to -X OR X. The explanation given to address 2) states that the SqRt of X^2 is positive, hence -X needs to be positive (which makes X negative).

So confused. What are the exceptions to the rule "SqRt of X^2 should equal |X| which is equal to -X OR X"?

Every positive number x has two square roots: √x, which is positive, and -√x, which is negative. Together, these two roots are denoted ±√x. The positive square root, √x is known as the principal square root of x.

By definition √x is positive.
Note that the positive or negative sign comes before the √ sign. That's your clue.

Whenever the square root sign '√' is used it is used to mean the principal square root, i.e. the positive square root. Hence, square roots of 4 are ±√4, i.e. -2 and 2. But √4 is always equal to 2 NOT -2.

To remove ambiguities we use the modulus notation.
We write √(x²) = |x|, so that we always get the principal square root, i.e. the positive square root. Now, |x| is always positive. Hence,

• For x > 0 --> √(x²) = |x| = x > 0
  For x < 0 --> √(x²) = |x| = -x > 0

Thus we always get the positive square root.

Let's take two examples,
1. For x = 2 = √(x²) = √[(2)²] = |2| = 2
2. For x = -2 = √(x²) = √[(-2)²] = |-2| = 2

Hope this clears your confusions.

Kancz44 wrote:What is the value of x?

1) √(x^4) = 9
2) √(x^2) = -x

Statement 1: √(x^4) = |x^2| = 9 --> x = ±3

Not sufficient

Statement 2: √(x^2) = |x| = -x --> x is negative

Not sufficient

1 & 2 Together: x = 3

Sufficient

The correct answer is C.

1947 wrote:can you please explain what are real numbers.
would all integers be considered real number.

For GMAT, you don't need to know anything other than real numbers. All numbers used in GMAT are real numbers. Integers, fractions, decimals, irrational numbers... everything.

Numbers other than real numbers are know as imaginary numbers, which are... well, imaginary and hence not part of GMAT syllabus. If you want to know about them, go to this webpage.