Exponent Simplification HELP PLS

[Kaunteya]

If 27^(4x + 2) × 162^(-2x) × 36^x × 9^(6–2x) = 1, then what is the value of x?


-9

-6

3

6

9

Holy crap this is retarded. How in the world are you to solve this in around 2 minutes without f-ing up. I love the GMAT!!!!

HELP
Kaunteya

[joshi.komal]

Hi Kaunteya,

Start off by simplifying all the terms in to their primary factors and then equate the powers of exponents on both sides of equation.

27^(4x + 2) * 162^(-2x) * 36^x * 9^(6–2x) = 1
or (3)^4(4x + 2) * (2*81)^(-2x) * (6*6)^x * 3^2(6–2x) = 1
or (3)^4(4x + 2) * (2)^(-2x) * (3)^4(-2x) * (3*2)^2x × 3^2(6–2x) = 1
or (3)^4(4x + 2) * (2)^(-2x) * (3)^(-8x) * (3)^2x*(2)^2x × 3^2(6–2x) = 1

or (3)^[4(4x + 2) +(-2x)+ 2x+2(6–2x)]* (2)^[(-2x)+2x] = 1
or (3)^[16x +8 -8x + 2x+12–4x]* (2)^[0] = 1
or (3)^[ 6x + 20] = 1 ^ 0 (anything raised to 0 is one)

now compare the powers on both sides
6x+ 20 = 0
or x = -20/6
or x ~ -3


Cross check your question and answer choices? Either one of them is incorrect.

Thanks
Komal

[torontogmat.com]

First, you do not need to solve it in 2 minutes. If you prepare well, some of the earlier questions can be solved easily, allowing you to remain CALM when you see a monstrosity like this question!

When you see complicated exponents, it's very likely that there will be some way to simplify.

In this case, the original:

27^(4x + 2) × 162^(-2x) × 36^x × 9^(6–2x) = 1

becomes:

3^3 to the (4x + 2) * 2*3^4 to the (-2x) * 4*3^2 to the x * 3^2 to the (6-2x) = 3^0

the 2^-2x in the second term and the 2^2x in the 3rd term (2^2 to the x) cancel each other, leaving an equation with all terms being 3 raised to a power.

the result is 12x + 6 + (-8x) + 2x + 12 - 4x = 0, which yields

2x + 18 = 0, so x = -9.


If the math looks tough, there's going to be a trick! Be confident, and keep plugging away.

Good luck!
Tony

[Kaunteya]

Hey Tony, thanks for the help on that question. I forgot to raise the 2 to the (-2x). So I was having quite the problem. Your approach was a hell of a lot faster than mine. Good help.

Kaunteya