Extracting X

[cjs]

The MGMAT problem named "Extracting X" showed up on a CAT exam. However, I don't understand one particular piece of the answer explanation (piece causing confusion is noted at the bottom). Any help is appreciated. Thanks

Question:

If x and y are integers and

15^x + 15^(x+1)
----------------- = 15^y
4^y

What is the value of x?

A) 2 B) 3 C) 4 D) 5 E) Cannot be determined


Answer explanation:

(15^x + 15^(x+1)) = (15^y)(4^y)
[15^x + 15^x(15^1)] = (15^y)(4^y)
(15^x)(1 + 15) = (15^y)(4^y)
(15^x)(16) = (15^y)(4^y)
(3^x)(5^x)(24) = (3^y)(5^y)(22^y)

Since both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal.

2y = 4 so y = 2.
x = y so x = 2.

The correct answer is A.

In the above explanation, I understand how 15^(x+1) separates out to: (15^x)(15^1). 15^1 = 15, but I don't understand how 15^x = 1 before combining with 1 to be 16. Could this be explained further?

[shaan17]

Hi,

[15^x + 15^x(15^1)] = (15^y)(4^y) ->step 1
(15^x)(1 + 15) = (15^y)(4^y) ->step 2

In step 1 above, 15^x is taken out as a common factor to give step 2. If you multiply the LHS of the eq in step 2
you will get step 1. Hope this answers the question.

Just for the benefot of the other readers, the last step :
(3^x)(5^x)(24) = (3^y)(5^y)(22^y)
should be written as :
(3^x)(5^x)(2^4) = (3^y)(5^y)(2^ 2y) ie, 2 to the power of 2y, which is the same as 4 to the power of y.

Cheers
Shaan

[StaceyKoprince]

Very nice. If you're still unsure, try setting up a simpler, analogous situation.

15 + 15*3 is the same thing as 15(1+3). Pull a 15 out of each term to put in front and keep what's left in the parentheses.

[Guest]

Thank you both for your replies - this helps in understanding.

[StaceyKoprince]

You're welcome!