If a and b, are positive, what is the value of a + b ?

Question

(1)  (3^a) (3^b)=81

(2)  (3^a) (5^b)=225

u1983 · Answer

(1)  (3^a) (3^b) =81 -  Sufficient - as  (3^a) (3^b) = 3^{a+b} = 3^4  ==>a+b=4
(2)  (3^a) (5^b) =225 -  Not Sufficient  - remember the cond is only positive not positive integer.
One way to think of the proof is :
225= 3.3.5.5 = 5.45
Case 1:  3^2  *  5^2  ==>a=2 & b=2.Hence a+b=4
Case 2:one component is 5 so b=1
Now 45 could be  3^a  ==>a=log45/log3 = 3.465 (  3^{3.465}  almost=45). Hence a+b=4.465

Hence Ans is A.

soloyolodolo · Answer

GMAT will not give false information between the two facts so if

1.) A+B = 4, then

2.) A+B = 4 as well. and in your explanation A+B = 4.465.

Wouldn't the correct solution be D?

2.) A = 2, B = 2?

u1983 · Answer

Hello  soloyolodolo  .... my sincere apologies. I really could not follow your point. It would be great if you can elaborate a little more.
Thanks in advance.

soloyolodolo · Answer

Gmat will never contradict.

In your explanation.

1.) A+B = 4

2.) A+B = 4.465

Does that make sense?

I agree with your explanation of the answer, however based on GMAT rules. I think answer would be D?

Maybe someone else can help. I know there are numerous people in the forums who are much more intelligent and math wizards.

Posted from my mobile device

KanishkM · Answer

Till the time someone else pitches in, inline is the  Official Explanation

Statement (1) gives you that  (3^a)(3^b)=81 
Remember that to find the product of two exponential expressions with a common base, you simply add their exponents.
That means that  (3^a)(3^b)=81  becomes 3^(a+b)=81
 
The right side of the equation can then be rewritten as  3^4  giving you the equation 3^(a+b) =  3^4 
so a+b=4

You can therefore conclude that statement (1) is sufficient when taken on its own.

Statement (2) gives that  (3^a)(5^b)=225 
If you factor 225, it becomes  (3^2)(5^2)=(3^a)(5^b) , At this point, it is tempting to say that a+b=4
But remember -- you haven’t been told even whether a and b are integers. 
It is possible to say that a=4 and that b is equal to an irrational number (log5  \frac{225}{81} , if you do the math).

Because the equation could yield multiple values of a and b
 
You must conclude that statement (2) is insufficient and choose answer choice (A).

bebs · Answer

soloyolodolo  I agree with you that the two statements would not contradict each other on GMAT's official questions. I also went with D.

bebs · Answer

Bunuel  Please I need help with this question. I got D but OA is A. Thanks in advance.

Bunuel · Answer

If a and b, are positive, what is the value of a + b ?

Notice that we are not told that a and b are integers.

(1) \((3^a) (3^b)=81\);

\(3^{a+b}=3^4\);

\(a + b = 4\).

Sufficient.

(2) \((3^a) (5^b)=225\).

Now, if were told that a and b are positive integers, then yes, from \((3^a) (5^b)=3^2*5^2\), it would follow that a = 2 and b = 2. But we are not given that, thus it's possible that a is say 1 and b is some irrational number (satisfying 5^b = 225 --> b = ~3.3652...). Not sufficient.

Answer: A.

Similar questions to practice:
https://gmatclub.com/forum/if-3-a-4-b-c- ... 06047.html
https://gmatclub.com/forum/if-x-2y-3-200 ... 92486.html
https://gmatclub.com/forum/if-x-3y-4-5- ... 76981.html

Hope it helps.

RupamPaul13 · Answer

If we consider any value of a and b other than a=2 and b=2 then 3^a*5^b will never become perfect 225 and only via approximation we need to conclude the value as 225 (a or b will be irrational number). So whether GMAT may test on such definite vs approximate results in real test?  Bunuel   chetan2u

Bunuel · Answer

You are wrong.

5^b = 225 has a solution for b, b = log(225)/log(5) (which is irrational number). For that value, 5^b is EXACTLY 225.

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If a and b, are positive, what is the value of a + b ?

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