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New post 12 Apr 2017, 06:05
stan3544 wrote:
Bunuel wrote:
stan3544 wrote:
And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?


Not sure I understand what you mean but GMAT doe not have its own math. That's simply not correct. The only thing is that GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers, so no imaginary numbers. That's why the even roots from negative numbers are not defined for the GMAT.

As for your example:

(-2)^4 = 16 but -2^4 = -16, this is true generally not only for the GMAT.


Hey Bunuel,

Now i am totally confused. Yes it is obvious and universally understood that sqr.root of negative number is not a real number, basically it doesn't exist, why? Because no negative number raised to an even power will ever give us a negative result, yet according to your example -2^4 = -16, so we know that the number -2 raised to the power 4 gives us -16, isn't it a paradox? Now, according to the exponent rules: (-2)^4 = (-2^1)^4 = -2*(-2)*(-2)*(-2) right? Now according to your interpretation -2^4 is not the same, so it basically means -2*2*2*2, which doesn't make sense, because this goes against the whole idea behind exponents, which is: we take a number and multiply by ITSELF n times. So the correct way to write it would be -(2^4) = -16 or -(-2^4) = -16. Now back to the issue at hand: logically 2^2=4 as well as -2*(-2)=4, which is (-2)^2= 4, now i don't see why sqr.root of 4 can only be 2 and not -2 as well.


-2^4 = -(2^4) = -16. It's order of operations thing.

As for the square roots: I tried to explain it several times on previous pages but I'll try this one last time:

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. Even roots have only a positive value on the GMAT.

In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5.

If it's still unclear or confusing then probably it's better to start from basics and brush-up fundamentals or enrol into a class.
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New post 30 May 2017, 23:18
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stan3544 wrote:
And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?


The gmat doesn't have its own version of math. The square root is a function, that's why it is always positive. As a function, it can't take a positive and a negative value (if it did, it wouldn't be a function, you can look at the graph to check that out).

When you solve x^2 = 4, applying the square root function and assuming two results for x is incorrect, because you would only get x = 2. The formal and correct method is:
x^2 = 4
x^2 - 4 =0
(x + 2)(x - 2)=0 and from here get both results for x.
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New post 30 May 2017, 23:21
CristianJuarez wrote:
stan3544 wrote:
And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?


The gmat doesn't have its own version of math. The square root is a function, that's why it is always positive. As a function, it can't take a positive and a negative value (if it did, it wouldn't be a function, you can look at the graph to check that out).

When you solve x^2 = 4, applying the square root function and assuming two results for x is incorrect, because you would only get x = 2. The formal and correct method is:
x^2 = 4
x^2 - 4 =0
(x + 2)(x - 2)=0 and from here get both results for x.


That's correct. + 1.

Another way of solving x^2 = 4 would be:

\(x = \sqrt{4}=2\) or \(x = -\sqrt{4}=-2\).
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New post 31 May 2017, 09:13
Can someone please provide the logic behind not taking the negative root for '√9=3, NOT +3 or -3;'
Is this mentioned anywhere int the OG or any other official material?
How do we know the GMAT doesn't consider the negative root ?
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New post 31 May 2017, 10:20
AllenF wrote:
Can someone please provide the logic behind not taking the negative root for '√9=3, NOT +3 or -3;'
Is this mentioned anywhere int the OG or any other official material?
How do we know the GMAT doesn't consider the negative root ?


This is explained many times. \(\sqrt{}\) denotes a function. Mathematically the square root function cannot give negative result.

OFFICIAL GUIDE:
\(\sqrt{n}\) denotes the positive number whose square is n.

Piece of advice, people here, experts/tutors, are here to help students, not to deceive them. So, if you see the same thing repeated many times by experts/tutors you should now that it's true.
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New post 31 May 2017, 19:33
Bunuel wrote:
CristianJuarez wrote:
stan3544 wrote:
And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?


The gmat doesn't have its own version of math. The square root is a function, that's why it is always positive. As a function, it can't take a positive and a negative value (if it did, it wouldn't be a function, you can look at the graph to check that out).

When you solve x^2 = 4, applying the square root function and assuming two results for x is incorrect, because you would only get x = 2. The formal and correct method is:
x^2 = 4
x^2 - 4 =0
(x + 2)(x - 2)=0 and from here get both results for x.


That's correct. + 1.

Another way of solving x^2 = 4 would be:

\(x = \sqrt{4}=2\) or \(x = -\sqrt{4}=-2\).


Yes, actually I'd like to correct myself, I hope it isn't more confusing.

Bunuel's solution applying square root function comes from the fact that the function f(x)=x^2 allows x to take positive or negative values (or 0):

I don't have 5 posts so I can't include the image :( , please search for it on google.

Meanwhile, the function \(f(x)= \sqrt{x}\) doesn't (this is why square root is always positive):

(check image on google)

So, if people want to solve this excercise by applying square root function, you must consider that this is incorrect:

\(x^2=4\) Apply \(\sqrt{}\)
\(\sqrt{x^2}=\sqrt{4}\)
\(x = \sqrt{4}\)
\(x=2\) or \(x= -2\)

It is incorrect because \(\sqrt{4}\) is always 2 and you lost a solution in the process.

So, you have to consider that mathemathically, \(\sqrt{x^2} = |x|\). This is only because \(f(x)=x^2\) can take a positive or a negative value for x (or 0 of course). Indeed, this doesn't mean the square root takes a positive or a negative solution, its always positive (or zero)!! (remember, \(|x|\) is always \(>= 0\) for any value of \(x\)).

You can check it out like this:
\(\sqrt{4} = \sqrt{2^2}= |2|=2, not -2!!\) or:
\(\sqrt{4} = \sqrt{(-2)^2}= |-2|=2, not -2!!\) It's impossible to get to \(-2\) like that

So the correct process by applying square root function is:
\(x^2=4\) Apply \(\sqrt{}\)
\(\sqrt{x^2}=\sqrt{4}\)
\(|x| = 2\)
\(x=2\) or \(x= -2\)
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New post 31 Dec 2017, 10:14
very good question. Definitely eye opener.
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New post 03 Feb 2018, 14:24
I love this question.

Did anyone come across any similar ones in the forum question banks? I'm not really able to drill down enough on filters to find similar ones with any consistency.
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New post 29 Jun 2018, 01:23
I think this is a poor-quality question and I don't agree with the explanation. the solution of minus two is correct , you can have the 4th root of (X=-2) it is a 4th root of 16 ,and 16 it is a positive number which can use for the root.
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New post 29 Jun 2018, 01:30
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Eyaltau wrote:
I think this is a poor-quality question and I don't agree with the explanation. the solution of minus two is correct , you can have the 4th root of (X=-2) it is a 4th root of 16 ,and 16 it is a positive number which can use for the root.


You are wrong. Your doubt is addressed MANY times on previous pages:

\(\sqrt{}\) denotes a function. Mathematically the square root function cannot give negative result. 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, +5 and -5. Even roots have only a positive value on the GMAT.

OFFICIAL GUIDE:
\(\sqrt{n}\) denotes the positive number whose square is n.

Piece of advice, people here, experts/tutors, are here to help students, not to deceive them. So, if you see the same thing repeated many times by experts/tutors you should now that it's true.

P.S. Generally I would suggest to brush-up fundamental before attempting hard questions.
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New post 02 Aug 2019, 21:10
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Can't we just plug-in the solutions back to the equation to test whether it works? Bunuel - can you please validate this approach?

power of 4 of LHS
\(x^4 = x^3 + 6x^2\)\(x^4 - x^3+6x^2 = 0\)
\(x^2(x^2 - x - 6)= 0\)\(x^2 (x-3)(x+2)=0\)
\(x=3, x=-2\) and \(x=0\)
Test valid solutions by plugging back in the roots.

\(x=-2\)
\((-2)^2((-2)^2-(-2)-6)\)\(= 4(4 + 4-6)=0\)
4(2) =8 and is not equal to 0 therefore -2 is not a valid solution

\(x=3\)
\((3)^2(3^2 - 3-6)\)
\(9(9-3-6)= 9(0) =0\)therefore valid.

Only 3 is a valid solution
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New post 22 Sep 2019, 23:15
I think this is a high-quality question and I agree with explanation.
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New post 22 Sep 2019, 23:23
dcummins wrote:
Can't we just plug-in the solutions back to the equation to test whether it works? Bunuel - can you please validate this approach?

power of 4 of LHS
\(x^4 = x^3 + 6x^2\)\(x^4 - x^3+6x^2 = 0\)
\(x^2(x^2 - x - 6)= 0\)\(x^2 (x-3)(x+2)=0\)
\(x=3, x=-2\) and \(x=0\)
Test valid solutions by plugging back in the roots.

\(x=-2\)
\((-2)^2((-2)^2-(-2)-6)\)\(= 4(4 + 4-6)=0\)
4(2) =8 and is not equal to 0 therefore -2 is not a valid solution

\(x=3\)
\((3)^2(3^2 - 3-6)\)
\(9(9-3-6)= 9(0) =0\)therefore valid.

Only 3 is a valid solution


Yes, that would be a valid approach. Solving, then verifying.
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New post 28 Oct 2019, 23:20
Hi Bunuel,

I tried to use vieta's theorem

\(x_1+x_2=\frac{-b}{a}\)

But now I realise that the theorem itself does not guarantee that the solution is not extraneous, right?

Could you confirm that?
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New post 28 Oct 2019, 23:23
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SchruteDwight wrote:
Hi Bunuel,

I tried to use vieta's theorem

\(x_1+x_2=\frac{-b}{a}\)

But now I realise that the theorem itself does not guarantee that the solution is not extraneous, right?

Could you confirm that?


Yes because you used it for the quadratics which you got AFTER you squared the whole expression, which (squaring) created an extra, false root.
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New post 28 Oct 2019, 23:53
Bunuel wrote:
SchruteDwight wrote:
Hi Bunuel,

I tried to use vieta's theorem

\(x_1+x_2=\frac{-b}{a}\)

But now I realise that the theorem itself does not guarantee that the solution is not extraneous, right?

Could you confirm that?


Yes because you used it for the quadratics which you got AFTER you squared the whole expression, which (squaring) created an extra, false root.


Since squaring is a non-reversible operation. Makes sense, thank you!
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Re: D01-13   [#permalink] 28 Oct 2019, 23:53

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