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If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?

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If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post Updated on: 17 May 2016, 21:48
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If \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6=2000\), what is the value of ab?

(1) a = 5
(2) a and b are positive integers

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Originally posted by madhavsrinivas on 23 Dec 2013, 11:18.
Last edited by Bunuel on 17 May 2016, 21:48, edited 2 times in total.
Edited the question.
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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 23 Dec 2013, 19:09
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madhavsrinivas wrote:
If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?

1) a = 5
2) a and b are positive integers

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked
C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
D) EACH statement ALONE is sufficient to answer the question asked
E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed


({a^1/2}*{b^1/3})^6=2000

==>a^3*b^2 = 2000

from statement 1 :

a = 5

125*b^2 = 2000

b^2 == 2000/125
b^2 == 16
b =4,-4

So ab will be 20,-20.

from statement 2

a,b are +ve

a^3*b^2 = 2000

this will be factorise in above only -- 125*16

so answer from this will be only 20.

So answer will be B.

Thanks
AB

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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 24 Dec 2013, 06:49
shouldn't the answer be C, considering the fact that it hasn't been given that a & b are integers? there can be endless values of a&b satisfying the second statement.
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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 24 Dec 2013, 11:37
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asethi100 wrote:
shouldn't the answer be C, considering the fact that it hasn't been given that a & b are integers? there can be endless values of a&b satisfying the second statement.

Hi asethi,

Simplifying the original equation gives us a^3 * b^2 = 2000
We can write 2000 as 125 * 16 or 1000 *2 or 40 * 50 etc etc.
But if you see statement 2, which says a and b are integers then only 125 and 16 can be expressed in the form of a^3 and b^2, i.e 5^3 and 4^2.
The other values such as 1000*2 or 40*50 can be expressed in the form of a^3 and b^2, but either of a and b or both, will not be of integer values.
So, if you consider statement 2 alone, you will get the answer straight away. I hope this helps ! :)
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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 28 Apr 2017, 21:10
madhavsrinivas wrote:
asethi100 wrote:
shouldn't the answer be C, considering the fact that it hasn't been given that a & b are integers? there can be endless values of a&b satisfying the second statement.

Hi asethi,

Simplifying the original equation gives us a^3 * b^2 = 2000
We can write 2000 as 125 * 16 or 1000 *2 or 40 * 50 etc etc.
But if you see statement 2, which says a and b are integers then only 125 and 16 can be expressed in the form of a^3 and b^2, i.e 5^3 and 4^2.
The other values such as 1000*2 or 40*50 can be expressed in the form of a^3 and b^2, but either of a and b or both, will not be of integer values.
So, if you consider statement 2 alone, you will get the answer straight away. I hope this helps ! :)


This is helpful. Is there any way to concretely prove that only a and b can only take the values of 5 and 4 respectively?
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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 29 Apr 2017, 07:22
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madhavsrinivas wrote:
If \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6=2000\), what is the value of ab?

(1) a = 5
(2) a and b are positive integers


Target question: What is the value of ab?

Given: \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6=2000\)
Simplify to get: (a³)(b²) = 2000

Statement 1: a = 5
Take (a³)(b²) = 2000 and replace a with 5 to get: (5³)(b²) = 2000
Simplify: (125)(b²) = 2000
Divide both sides by 125 to get: b² = 16
So, EITHER b = 4 OR b = -4
Case a: if b = 4, then ab = (5)(4) = 20
Case b: if b = -4, then ab = (5)(-4) = -20
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a and b are positive integers
We're told that (a³)(b²) = 2000
We also know that 2000 = (5)(5)(5)(2)(2)(2)(2) = (5)(5)(5)(4)(4) = (5³)(4²)
Since we're told that a and b are positive integers, we can conclude that a = 5 and b = 4, which means ab = (5)(4) = 20
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

Cheers,
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If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 24 May 2017, 12:04
My approach, in case it helps:

Question stem: \((a^{1/2}*b^{1/3})^6=2000\)

Simplify it first:

\({(a^{1/2})}^{6}*{(b^{1/3})}^{6}=2000 \rightarrow a^{3}*b^{2}=2000\)

Now do the prime factorization of \(2000\) to see that \(2000=2^{4}*5^{3}\), which we can transform, using exponent properties, to \(2000=4^{2}*5^{3}\). Now we have an expression similar to that of the question stem. Therefore, \(a\) should be \(5\).

From here, we should pay attention to the odd/even nature of the exponents. For base \(5\), the exponent is odd (\(3\)). Since \(2000\) is positive and the exponent of base \(4\) is even (\(2\)), we know that \(5\) must be positive. However, since the exponent of \(4\) is even, \(4\) could actually be \(4\) or \(-4\). The output of \(-4^{2}\) is the same as of \(4^{2}\), which is \(16\). Therefore, \(b\) could be \(4\) or \(-4\).

So we must focus on the sign of that \(4\), or in the problem's language, the sign of \(b\).

Statement 1) \(a=5\). We already knew this. Not sufficient.

Statement 2) \(a\) and \(b\) are positive integers. We know that \(a=5\) and that \(b\) is positive, so \(b\) is \(4\). From here, we know that \(a*b=20\). Sufficient.

Hope this approach helps.
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If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 17 Sep 2017, 21:22
0 → see question
1 → \(a^3 * b^2 = 2000\)
2 → \((ab)^2 * a = 2000\)
3 → \((ab)^2 = 2000 / a\)

Statement #1: \(a = 5\)
From 3 → \((ab)^2 = 2000 / 5 = 400\)
So \(ab = \sqrt{(400)} = ± 20\), but GMAT doesn't like ± (unlike my grade school algebra teacher)
So Statement #1 is insufficient.

Statement #2: a and b are positive integers
From 3 → \((ab)^2 = 2000 / a\)
So \(ab = \sqrt{(2000 / a)} = 2 * 2 * 5 * \sqrt{(5 / a)} → a = 5, b = 2 * 2\) since both are integers > 0
So Statement #2 is sufficient

Since a = 5, can be determined from Statement #2 alone, I will answer B for BOOM
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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?  [#permalink]

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New post 02 Oct 2018, 10:50
Kardo wrote:
0 → see question
1 → \(a^3 * b^2 = 2000\)
2 → \((ab)^2 * a = 2000\)
3 → \((ab)^2 = 2000 / a\)

Statement #1: \(a = 5\)
From 3 → \((ab)^2 = 2000 / 5 = 400\)
So \(ab = \sqrt{(400)} = ± 20\), but GMAT doesn't like ± (unlike my grade school algebra teacher)
So Statement #1 is insufficient.

Statement #2: a and b are positive integers
From 3 → \((ab)^2 = 2000 / a\)
So \(ab = \sqrt{(2000 / a)} = 2 * 2 * 5 * \sqrt{(5 / a)} → a = 5, b = 2 * 2\) since both are integers > 0
So Statement #2 is sufficient

Since a = 5, can be determined from Statement #2 alone, I will answer B for BOOM



Another approach to the same question:

Solution:

Simplifying the equation in the stimulus yields

\(a^3b^2\)=2000

Quote:
Statement (1) by itself is insufficient. If a = 5, then \(5^3b^2=2000\) or\(b^2=16\).

Thus b could equal 4 or -4, yielding two possible values for ab.


Quote:
Statement (2) by itself is sufficient.

If we were to break 2000 down into prime factors, we would get that 2000 = \(5^3.2^4\)

Since \(2^4=4^2\),

2000 can be rewritten as \(5^34^2\)

Thus, a = 5 and b = 4 and we can determine the value of ab.

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Re: If ({a^1/2}*{b^1/3})^6=2000, what is the value of ab?   [#permalink] 02 Oct 2018, 10:50
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