Problem Solving

#163 of the Official Quant Review (green):

(8^2)(3^3)(2^4)/(96^2)=

(8^2)(3^3)(2^4)/(8^2)(3^2)(2^4)= 3

Can someone explain how I could figure out that 96^2 is equal to (8^2)(3^2)(2^4)? I understand how to work the problem but not how I can figure out that such a large number (96^2) is broken down into smaller powers. Thanks in advance!

The principle is that you want to delete factors which are both at the denominator and at the numerator so you will try to break into pieces 96²

How to arrive to that: 96² = 8² * 3² * 2^4

96=2*48=2*2*24=2*2*2*12=2*2*2*2*6=2^5*3
so
96²=2^10*3²=2^4*2^6*3²=2^4*3²*8²

Whenever you see exponents like these always prime factorize.

You should remember all the powers of 2 till 10

2^1 =2
2^2 =4
2^3 =8 so on so forth.


(8^2)(3^3)(2^4)/(96^2)


96 = 3 * 2^5
96^2 = (3^2) (2^10)

8 = 2^3
8^2 = 2^6

Numerator (8^2)(3^3)(2^4) = (2^6)(3^3)(2^4) = (2^10)(3^3)
Denominator (3^2) (2^10)

(2^10)(3^3) / (3^2) (2^10) =3

Hope this helps.

Parallel that was the perfect explanation for my thought process. Thanks!