Question Bank - Equations, Inequalities, & VIC's

[aaa]

If the bases were not prime numbers but still equal to one another, could the problem be solved with the same methodology, ie setting the exponents equal to one another?

If (22x+1)(32y-1) = 8x27y, then x + y =


-3
-1
0
1
3
Let's rewrite the right side of the equation in base 2 and base 3: (22x+1)(32y-1) = (23)x(33)y. This can be rewritten as: (22x+1)(32y-1) = 23x33y

Since both bases on either side of the equation are prime, we can set the exponents of each respective base equal to one another:

2x + 1 = 3x, so x = 1
2y - 1 = 3y, so y = -1

Therefore, x + y = 1 + (-1) = 0.

The correct answer is C.

[StaceyKoprince]

Yes - that explanation should really say "since both bases are equal..." Can you tell me the specific question number so I can go edit the explanation? Thanks!

[aaa]

25

[StaceyKoprince]

Thanks! I had a further conversation with our curriculum director about this problem, and he said that it was worded that way because we were trying to make sure people realized that you have to make sure that you have different primes - in other words, that you have combined the bases that you can combine.

So, for example, if you had:
(4^x)(16^y) = (4^2)(16^3) this is NOT sufficient to say that x=2 and y=3, because 4 and 16 both break down to the same prime (2) and so you could have different integer combinations for x and y (try it on a calculator!).

So, the bases on each side do need to be equal in order to drop the bases and set the exponents equal to each other. In addition, if you have multiple bases, any of the same primes have to be combined. In the original example, we've combined all the 2s with each other and all of the 3s with each other. In the example I just typed earlier in this post, we haven't - both of those can break down into 2s. Make sense?

[guest]

(22x+1) was written to mean 2^(2x+1).... i get it now... When i just read the first post of this thread, it was impossible to make sense out of how the answer came out the way it did. ^ is a standard notation for exponent in scientific field of study (programming, engineering, etc. scientific calculator uses this rule). Also, just to note, (22x+1) is understood as 22 * x + 1, as standard (* is sign for multiplication).

thanks

[aaa]

makes sense. Thx

[rfernandez]

You're welcome.