For a negative integer x, what is the value of x*root(x^2)?

SOME NOTES:

1. GMAT is dealing only with Real Numbers: Integers, Fractions and Irrational Numbers.

2. Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, \(\sqrt{25}=+5\) and \(-\sqrt{25}=-5\). Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).

As for \(\sqrt{x}\) and \(x^{\frac{1}{2}}\): they are the same.

3. \(\sqrt{x^2}=|x|\).

The point here is that as square root function can not give negative result then \(\sqrt{some \ expression}\geq{0}\).

So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?

Let's consider following examples:
If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);
If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).

So we got that:
\(\sqrt{x^2}=x\), if \(x\geq{0}\);
\(\sqrt{x^2}=-x\), if \(x<0\).

What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).


BACK TO THE ORIGINAL QUESTION:
For a negative integer \(x\), what is the value of \(x*\sqrt{x^2}\)?
A. x^2
B. -x^2
C. x
D. -x
E. -1

Given: \(x<0\). Question: \(x*\sqrt{x^2}=?\)

According to the above notes: \(x*\sqrt{x^2}=x*|x|\), since \(x<0\) then \(|x|=-x\) --> \(x*|x|=x*(-x)=-x^2\).

Or just substitute some negative \(x\), let \(x=-2<0\) --> \((-2)*\sqrt{(-2)^2}=(-2)*2=-4=-(-2)^2\).

Answer: B.

Hope it's clear.

P.S. Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/ and DS questions in the DS subforum: gmat-data-sufficiency-ds-141/ No posting of PS/DS questions is allowed in the main Math forum.