If x not equal to 0, then (sqrt(x^2) )/x =

eitijan wrote:If x not equal to 0, then (√x²)/x =

A) -1
B) 0
C) 1
D) x
E) |x|/x

Case 1: x=2
In this case, (√x²)/x = (√2²)/2 = √4/2 = 2/2 = 1. This is our target.

Plugging x=2 into the answer choices, we get:
A) -1
B) 0
C) 1
D) x = 2
E) |x|/x = |1|/1 = 2/2 = 1

Only C and E yield our target value of 1.
Eliminate A, B and D.

Case 2: x=-2
In this case, (√x²)/x = (√(-2)²)/-2 = √4/-2 = 2/-2 = -1. This is our target.

Plugging x=-2 into the remaining answer choices, we get:
C) 1
E) |x|/x = |-2|/-2 = 2/-2 = -1

Only E yields our target value of -1.
Eliminate C.

The correct answer is E.

What is the flaw in below approach?

(x^2)^1/2=(x^2*1/2 )/x (by exponent rule) = x/x =1

The value in red is incorrect.
√(x²) ≠ x.

By definition:
√ means the POSITIVE ROOT ONLY.
Thus:
√(x²) = the POSITIVE ROOT of x² = |x|.

But why cant we apply exponent rule here i.e. (x^a)^b = x^a*b?

eitijan wrote:But why cant we apply exponent rule here i.e. (x^a)^b = x^a*b?

Because √ means the POSITIVE ROOT ONLY.

Thus:
√x ≠ x^(1/2).
To illustrate:
√4 = 2.
4^(1/2) = ±2.
Since √4 = 2 and 4^(1/2) = ±2, we cannot claim that √4 = 4^(1/2).

Correct:
√x = |x^(1/2)|.

eitijan wrote:But why cant we apply exponent rule here i.e. (x^a)^b = x^a*b?

This is a great question. The issue is that squaring x turns a negative into a positive, so we aren't working with x any more when we take the square root.

For instance, suppose x = -3. Then x² = 9, and √x² = 3. So we now have 3, which is really -x (i.e. -(-3)).

A simple way of escaping all this is to treat √x² as |x|, or to consider each case. For instance, if x > 0:

√x² = (x²)½ = x

and if x < 0

√x² = (-x²)½ = -x

So we'll end up with either x/x or -x/x, e.g. 1 or -1. The only answer that fits both cases is E.

Thanks for elaborating it.
I understood it now and my confusion got cleared.

eitijan wrote:If x not equal to 0, then (sqrt(x^2) )/x =

A) -1
B) 0
C) 1
D) x
E) |x|/x

Source : GmatPrep

What is the flaw in below approach?

(x^2)^1/2=(x^2*1/2 )/x (by exponent rule) = x/x =1
eg: (3^2*1/2)/3 = 3/3 = 1
if x = -ve, ((-3)^2*1/2 )/-3 = -3/-3 =1

So answer should be C. Please respond

Hi eitijan,

the flaw is in the above highlighted (blue) step

Concept: Square root of anything is ALWAYS POSITIVE and never negative

But here you don't know the sign of x which may be positive or negative both

hence, Sqrt(x) should be |x|

therefore, (sqrt(x^2) )/x = |x|/x = +1 for Positive values of x and -1 for Negative values of x

I hope this helps!!!

Answer: Option E

eitijan wrote:Thanks for elaborating it.
I understood it now and my confusion got cleared.

No prob, happy to help!