Problem Solving

If x != 0, then sqrt(x^2)/x =

1. -1
2. 0
3. 1
4. x
5. |x|/x

Source Practice Test.

airan wrote:If x != 0, then sqrt(x^2)/x =

1. -1
2. 0
3. 1
4. x
5. |x|/x

Source Practice Test.

Just for clarification, I assume that "!="means "is not equal to".

The key to this question is recognizing that the sqrt symbol literally translates as "the positive square root of". So, if you saw a question that asks:

What's the sqrt(25), the answer would be ONLY +5.

Since we translate the notation in this manner, we know that, no matter the sign of x, sqrt(x^2) is always going to be positive. In other words, the numerator in the expression will be the absolute value of x.

So, we end up with:

|x|/x ... choose (5)

Thnx Stuart ..
As per the explanation

sqrt(x^2) should always be x so we can say that sqrt(x^2)/x is same as x/x which amounts to 1.

airan wrote:Thnx Stuart ..
As per the explanation

sqrt(x^2) should always be x so we can say that sqrt(x^2)/x is same as x/x which amounts to 1.

That will only be correct if x is positive.

sqrt(x^2) will always be the absolute value of x.

For example, if x = -4, then:

sqrt(16) = +4 (which is NOT the same as x, which is -4).

Following through the entire expression:

sqrt(-4^2)/-4 = 4/-4 = -1

So, if x is positive, then the expression will simplify to +1; if x is negative, the expression will simplify to -1. Only choice (E), |x|/x, will always be correct.

this is a gmatprep problem and the OA is E: |x|/x

Stuart Kovinsky wrote:

airan wrote:If x != 0, then sqrt(x^2)/x =

1. -1
2. 0
3. 1
4. x
5. |x|/x

Source Practice Test.

Just for clarification, I assume that "!="means "is not equal to".

The key to this question is recognizing that the sqrt symbol literally translates as "the positive square root of". So, if you saw a question that asks:

What's the sqrt(25), the answer would be ONLY +5.

Since we translate the notation in this manner, we know that, no matter the sign of x, sqrt(x^2) is always going to be positive. In other words, the numerator in the expression will be the absolute value of x.

So, we end up with:

|x|/x ... choose (5)

Why are we assuming here only the positive square root? For GMAT purposes, isn't it insufficient if we reach squareroot something as an asnwer, and unless the negative root is automatically disqualified (given in answer stem, or dealing with geometry, etc), we would say this is insufficient to answer a question?
What is it here which is precluding us from considering the negative root? Especially since in the answer we have allowed for the posibility of x being a negative number by making it an absolute value.

The way i looked at this was to say if x is ultimately negative, then so will the denominator, thus leaving +1. and if x is positive, so will the denominator, and thus +1 again.

zank wrote:
Why are we assuming here only the positive square root? For GMAT purposes, isn't it insufficient if we reach squareroot something as an asnwer, and unless the negative root is automatically disqualified (given in answer stem, or dealing with geometry, etc), we would say this is insufficient to answer a question?
What is it here which is precluding us from considering the negative root? Especially since in the answer we have allowed for the posibility of x being a negative number by making it an absolute value.

The way i looked at this was to say if x is ultimately negative, then so will the denominator, thus leaving +1. and if x is positive, so will the denominator, and thus +1 again.

√x always returns a non-negative number for all non-negative values of x. For example √16=4. Only 4, not -4.

Now, it is true that 16 has two square roots: 4 and -4. We refer to 4 as the principal square root and -4 as the negative square root. However, when we use the symbol, √, we are referring to ONLY the principal square root of a number, which is always non-negative.

You can also think of it working the same way it does when you plug it into a calculator. When you ask a calculator what √16 is, it doesn't say "4 or -4", it just says 4.

Thanks pete. So if on the GMAT we reach an answer x^2=16, can we conclude then that x=4, and thus this would be enough to make a statement sufficient?

Alternatively if we reach an answer x^2=y^2, then can we also conclude that x=y, or in this case since we are dealing with variables then we cannot conclude that x=y?

GmatMathPro wrote:

zank wrote:
Why are we assuming here only the positive square root? For GMAT purposes, isn't it insufficient if we reach squareroot something as an asnwer, and unless the negative root is automatically disqualified (given in answer stem, or dealing with geometry, etc), we would say this is insufficient to answer a question?
What is it here which is precluding us from considering the negative root? Especially since in the answer we have allowed for the posibility of x being a negative number by making it an absolute value.

The way i looked at this was to say if x is ultimately negative, then so will the denominator, thus leaving +1. and if x is positive, so will the denominator, and thus +1 again.

√x always returns a non-negative number for all non-negative values of x. For example √16=4. Only 4, not -4.

Now, it is true that 16 has two square roots: 4 and -4. We refer to 4 as the principal square root and -4 as the negative square root. However, when we use the symbol, √, we are referring to ONLY the principal square root of a number, which is always non-negative.

You can also think of it working the same way it does when you plug it into a calculator. When you ask a calculator what √16 is, it doesn't say "4 or -4", it just says 4.

No, that's a totally different matter. In that case x=4 or x=-4. If x^2=y^2, then x=y or x=-y. The confusion may lie in the fact that people will informally tell you to solve x^2=16 by "taking the square root of both sides", but really you have to include the positive and negative square roots if you solve it that way because x=4 and x=-4 both satisfy the equation.

The solutions to x^2=16 are, by definition, the square roots of 16. Every positive number has two square roots: a principal non-negative square root, and a negative square root. The two square roots of 16 are 4 and -4, both of which satisfy this equation. But when we say √16 we are referring ONLY to 4, not -4. Just think of it as a notational convention. We're not excluding -4 because (-4)^2 doesn't equal 16. Obviously it does. We exclude it because we've defined the symbol '√' to refer only to the non-negative square root.

The avoid this confusion, you may wish to solve an equation like x^2=16 by the following method:

x^2=16
x^2-16=0
(x+4)(x-4)=0
x=4, x=-4

rather than "taking the square root of both sides", as this phrase is not quite precise enough.

zank wrote:Thanks pete. So if on the GMAT we reach an answer x^2=16, can we conclude then that x=4, and thus this would be enough to make a statement sufficient?

Alternatively if we reach an answer x^2=y^2, then can we also conclude that x=y, or in this case since we are dealing with variables then we cannot conclude that x=y?

GmatMathPro wrote:

zank wrote:
Why are we assuming here only the positive square root? For GMAT purposes, isn't it insufficient if we reach squareroot something as an asnwer, and unless the negative root is automatically disqualified (given in answer stem, or dealing with geometry, etc), we would say this is insufficient to answer a question?
What is it here which is precluding us from considering the negative root? Especially since in the answer we have allowed for the posibility of x being a negative number by making it an absolute value.

The way i looked at this was to say if x is ultimately negative, then so will the denominator, thus leaving +1. and if x is positive, so will the denominator, and thus +1 again.

√x always returns a non-negative number for all non-negative values of x. For example √16=4. Only 4, not -4.

Now, it is true that 16 has two square roots: 4 and -4. We refer to 4 as the principal square root and -4 as the negative square root. However, when we use the symbol, √, we are referring to ONLY the principal square root of a number, which is always non-negative.

You can also think of it working the same way it does when you plug it into a calculator. When you ask a calculator what √16 is, it doesn't say "4 or -4", it just says 4.