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

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Bunuel
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If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 02:27
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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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B
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 02:59
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{A^1/2*B^1/3}^6 = 2000
A^3*B^2 = 2000
A^3*B^2 = 2*2*2*2*5*5*5
A^3*B^2 = (±2)^4*5^3

Statement 1: A = 5
But B can be +4 or -4
(Insufficient)

Statement 2: a and b are positive.
A = 5 and B = 4
Therefore a*b = 4*5 = 20
(Sufficient)

Answer is B

Posted from my mobile device
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 03:52
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Bunuel 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


Solution


Step 1: Analyse Question Stem


    • \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6 = 2000\)
    • \(⟹ a^{(\frac{6}{2})}*b^{(\frac{6}{3})} = 2000\)
    • \( ⟹ a^3*b^2 = 125*16\)
We need to find the value of a*b

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: a = 5
    • So, \(5^3*b^2 = 125*16⟹ b^2 =16 ⟹ b = 4\) or \(-4\)
    • So, a*b can be either 20 or -20
    • We are getting contradictory results.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: a and b are positive integers
    • Since, a and b are positive,
      o \(a^3*b^2 = 125*16⟹a^3*b^2 = 5^3*4^2 \)
      o so a = 5 and b = 4
      o Thus, \(a*b = 5*4 = 20\)
Hence, statement 2 is sufficient.
Thus, the correct answer is Option B.
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 04:46
Kindly see the attachment.

Statement 2 is basically the hint for negating Statement 1

IMO B
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 05:22
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Bunuel 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


Given, \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6=2000\)

ie.. \((a^3*b^2)=2^4*5^3\)

ie.. \((a^3*b^2)=4^2*5^3\)

Question: a*b = ?

Statement 1: a = 5

Since, \((a^3*b^2)=4^2*5^3\)
therefore, b^2 = 4^2
i.e. b = +4 or -4 hence
ab = +20 or -20 hence

NOT SUFFICIENT

STatement 2: a and b are positive integers

Also, Since, \((a^3*b^2)=4^2*5^3\)

ie.. a = 5 and b = 4
i.e. ab = +20

SUFFICIENT

Answer: Option B

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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 07:49
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yashikaaggarwal wrote:
{A^1/2*B^1/3}^6 = 2000
A^3*B^2 = 2000
A^3*B^2 = 2*2*2*2*5*5*5
A^3*B^2 = (±2)^4*5^3

Statement 1: A = 5
But B can be +2 or -2
(Insufficient)

Statement 2: a and b are positive.
A = 5 and B = 4
Therefore a*b = 4*5 = 20
(Sufficient)

Answer is B

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yashikaaggarwal

I think you meant +-4.
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   03 Aug 2020, 08:37
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What is ab=?
Simplifying the given expression, we have:

a^3*b^2 = 2000 = 2 * 10^3 = 2 * (2^3*5^3) = 5^3 * 2^4 = 5^3*4^2

Therefore, comparing we get:

a=5 ; b = +4 or -4

Statements:

(1) a = 5
We don't know anything about b. ab can be 20 or -20.
Insufficient

(2) a and b are positive integers
We know for sure that a = 5 and (2) gives us b = 4.
Therefore, ab = 20.
Sufficient

Hence, the answer is Option(B).
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   29 Oct 2020, 04:02
If \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6=2000\), what is the value of ab?

= \(a^3b^2 = 2^45^3\)
= \(a^3b^2 = 4^25^3\)


(1) a = 5

This statement tells us a = 5.
Thus \(a^3\) = 125.
Thus \(b^ 2\) = 16

However, we don't know if b is 4 or -4.

If b = -4 then 5 * (-4) = -20
If b = 4 then 5 * 4 = 20

Insufficient.

(2) a and b are positive integers

= \(a^3b^2 = 4^25^3\)

Thus a = 5; b = 4
ab = 20.

Sufficient.

Answer is B.
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   29 Oct 2020, 10:54
Hello,
Ain't the answer should be C ?
Since we are using a=5 (from statement 1) and through that we getting the value of b as 4 (as a and b are positive integers) and hence the final value of ab ?
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   29 Oct 2020, 11:23
sagar23691 wrote:
Hello,
Ain't the answer should be C ?
Since we are using a=5 (from statement 1) and through that we getting the value of b as 4 (as a and b are positive integers) and hence the final value of ab ?


We can determine that a = 5 from statement 2 because we're told a and b are positive integers.

The factorization is \(4^25^3\). There is no other possibility for a and b since we're told they are positive integers.
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   29 Oct 2020, 11:41
GMATWhizTeam wrote:
Bunuel 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


Solution


Step 1: Analyse Question Stem


    • \((a^{(\frac{1}{2})}*b^{(\frac{1}{3})})^6 = 2000\)
    • \(⟹ a^{(\frac{6}{2})}*b^{(\frac{6}{3})} = 2000\)
    • \( ⟹ a^3*b^2 = 125*16\)
We need to find the value of a*b

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE


Statement 1: a = 5
    • So, \(5^3*b^2 = 125*16⟹ b^2 =16 ⟹ b = 4\) or \(-4\)
    • So, a*b can be either 20 or -20
    • We are getting contradictory results.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: a and b are positive integers
    • Since, a and b are positive,
      o \(a^3*b^2 = 125*16⟹a^3*b^2 = 5^3*4^2 \)
      o so a = 5 and b = 4
      o Thus, \(a*b = 5*4 = 20\)
Hence, statement 2 is sufficient.
Thus, the correct answer is Option B.


Great explanation.
thank you.
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Re: If (a^(1/2)*b^(1/3))^6 = 2000, what is the value of ab? [#permalink] New post   30 Jan 2022, 07:37
Official Explanation: Simplifying the equation in the stimulus yields a^3b^2=2000

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.

Statement (2) by itself is sufficient. If we were to break 2000 down into prime factors, we would get that 2000 = 5^32^4. 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] New post   02 Jun 2023, 05:34
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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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