gmat prep question - exponent

x= 15

2^x-2^x-2=3(2^13)

= 2^x - 2^x 2 ^ -2 = 3 (2^13) {splitting 2^x-2 as 2 ^ x * 2 ^-2 since a^m a ^n = a ^ m+n here m =x n = -2}

2^x (1 - 1/4) = 3 (2^13) {2 ^ -2 = 1/4 since a ^ -n = 1/a ^n}

2^x * 3/4 = 3 (2^13)

2^x * 3 /2 ^ 2 = 3 (2^13) {4 = 2 ^ 2}

2^x-2 . 3 = 3 ( 2^13)
{2 ^ x / 2^ 2 = 2 ^ x-2
using the rule a ^ m / a ^ n = a ^ m-n here m = x n = 2 }

x - 2 = 13

x =15

If I am not mistaken this problem has been solved in the forum but providing the solution again since I cant remember the link subject off the top of my head.

The key to problems like these like is to get the side where x is to the like bases/exponents to match the other side

Hope this helps!

positive !
this helps a lot !

could you explain a little further on this:
2^x (1 - 1/4)

the reason for 2^x - 2^(x-2) written as 2^x ( 1- 1/4) is:
{ plugging the values }

2^4 - 2^2 = 16- 4 = 12 or
2^4 ( 1- 1/4) = 16 ( 3/4) = 12
or
2^5 - 2^3 = 2^5 ( 1- 1/4) = 32 (3/4) = 24

i hope, it is clear now

good luck!

robzoc wrote:2^x-2^x-2=3(2^13)

can some one explain how to solve this ?

thank you

whenever dealing with exponents you always want to factor out the smaller value first, makes life a LOT easier...

2^x - 2^x-2 = 3(2^13)
(factor out 2^x-2 because its the smaller of the two)
2^x-2{2^2 -1} = 3(2^13)
2^x-2{4-1} = 3(2^13)
2^x-2{3) = 3(2^13)
(divide each side by 3 to cancel them out)
2^x-2 = 2^13
x-2 = 13
x = 15

Hope that helps, let me know if you have any questions

Splitting 2^x into 2^(x-2) * 2^2 is the key step here.