{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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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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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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yashikaaggarwal

I think you meant +-4.

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).

Kindly see the attachment.

Statement 2 is basically the hint for negating Statement 1

IMO B

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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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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Great explanation.
thank you.