If a^(2/3) = b^(2/3) for a ≠ 0 and b ≠ 0, then which of the following

Question

If  a^{\frac{2}{3}}=b^{\frac{2}{3}}  for  a ≠ 0  and  b ≠ 0 , then which of the following statements must be true?

I.  (\frac{a}{b})^2=1

II.  a^2=b^2

III.  \sqrt{a}=\sqrt{b}

A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III

Kinshook · Accepted Answer

Asked: If  a^{\frac{2}{3}}=b^{\frac{2}{3}}  for  a ≠ 0  and  b ≠ 0 , then which of the following statements must be true?

Cubing both sides
 a^2 = b^2 
 \frac{a^2}{b^2} = (\frac{a}{b})^2 = 1

I.  (\frac{a}{b})^2=1 
TRUE

II.  a^2=b^2 
TRUE

III.  \sqrt{a}=\sqrt{b} 
a = b or a = - b
NOT NECESSARILY TRUE

IMO C

rajatchopra1994 · Answer

Explanation:

a^2/3=b^2/3
means, (a/b)^2/3 = 1
raise to power multiply both sides by 3
(a/b)^2 = 1^3
(a/b)^2 = 1 (which is I)
a^2 = b^2 (which is II)

III. √a = √b ; both can be positive signs or opposite signs No.

IMO-C

Harsh9676 · Answer

IMO C

a^(2/3) = b^(2/3) can be written as (a^(1/3))^2 = (b^(1/3))^2.

Rule: 1. (a^m)^n = a^(mn)
 2. a number with an even power is always positive. However, a, b can either be positive or negative.

Using Rule 2 , Only St 1 and St 2 will satisfy.

Thanks
Harsh

Akhilesh1227 · Answer

Hi,

Whenever we use square root on variable, we consider the +ve root so why for III sentence -ve root is also considered in solutions?

Krunaal · Answer

The rule is that, where  a  is  non-negative ,  \sqrt{a}  will always be positive, for eg.  \sqrt{9} = 3 , it cannot be  -3 . In the rule you're referring to,  we are given   a   to be a non-negative integer , hence the result of that is positive. If  a  were negative, then the result of  \sqrt{a}  is an imaginary number.

In the question asked,  we don't know the signs  of  a  and  b , so if  b  is negative, and  a  is positive, then  \sqrt{b}  will be an imaginary number, and will not be equal to  \sqrt{a} . Hope it helps.

Akhilesh1227 · Answer

Hi krunaal

Thanks for the explanation

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If a^(2/3) = b^(2/3) for a ≠ 0 and b ≠ 0, then which of the following

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If a^(2/3) = b^(2/3) for a ≠ 0 and b ≠ 0, then which of the following

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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Perfect 805 Score Debrief by Julia

My Rewards