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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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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

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

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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.

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

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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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02 May 2015, 07:12

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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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17 May 2016, 12:28

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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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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?

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

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.

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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