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22 Jan 2012, 23:17
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In one Site I read the below fact. How true is it. Does Gmat provide this special condition.
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
23 Jan 2012, 04:23
Kudos
Bookmarks
sujit2k7
In one Site I read the below fact. How true is it. Does Gmat provide this special condition.
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
Q. For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
Q. For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
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.
General Discussion
manalq8
Joined: 12 Apr 2011
Last visit: 12 Feb 2012
Posts: 114
Own Kudos:
Given Kudos: 52
Status:D-Day is on February 10th. and I am not stressed
Affiliations: American Management association, American Association of financial accountants
Location: Kuwait
Concentration: finance and international business
Schools:Columbia university
Q18 V17 GMAT 2: 320 Q18 V19 GMAT 3: 620 Q42 V33
GPA: 3.48
23 Jan 2012, 07:43
Kudos
Bookmarks
Bunuel
sujit2k7
In one Site I read the below fact. How true is it. Does Gmat provide this special condition.
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
Q. For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
√x : √ sign (the radical sign) is called the "Principal root" and always results in a non-negative number.
(x)^1/2: The x to the power of 1/2 is the result of the equation, y^2 = x. In this case the result could be "positive or negative or zero".
So here is the conclusion
√x : Always non-negative (note I am not saying its always "positive")
(x)^1/2: Can be positive or negative or zero
Q. For a negative integer x, what is the value of x * √(x^2) ?
A) x^2
B) -x^2
C) x
D) -x
E) -1
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 \(x=-2<0\) --> \([m](-2)*\sqrt{(-2)^2}=(-2)*2=-4=-(-2)^2\).
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.
Banuel, I have a question.
why can't we cancel out the square root with the power of 2?? let me show u!
x * √(x^2)
we can do X* (sqrt x)^2, then we will be able to cancel the sqrt with power and we end up with x^2 which is ans A?
your explain is greatly appreciated. thanks
