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

Question

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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yashikaaggarwal · Answer

{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

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GMATWhizTeam · Answer

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.

sambitspm · Answer

Kindly see the attachment.

Statement 2 is basically the hint for negating Statement 1

IMO B

GMATinsight · Answer

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

KaramveerBakshi · Answer

yashikaaggarwal

I think you meant +-4.

KaramveerBakshi · Answer

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).

Basshead · Answer

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.

sagar23691 · Answer

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 ?

Basshead · Answer

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.

YASH750 · Answer

Great explanation. 
thank you.

rvgmat12 · Answer

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

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