Bunuel wrote:
If x and y are integers and \(72^x∗54^y=96\), what is the value of x−y?
A. -1
B. 0
C. 1
D. 2
E. 3
My reasoning:
72 broken down into primes is \(2^3 * 3^2\) After multiplying
x in we get \({2^3x} * 3^{2x}\)
54 broken down into primes is \(2 * 3^3\) After multiplying
y in we get \(2^y * 3^{3y}\)
96 broken down into primes is \(2^5 * 3\)
Same base means you can add the exponents together so we get: \(2^{(3x+y)} * 3^{(2x+3y)}\)
We know \(2^{(3x+y)} = 2^5\) and \(3^{(2x+3y)} = 3^1\)
We can solve for x and y now which gives us 2 and -1 respectively. Subtract them and we get 3 (Choice E)