If x and y are positive integers such that 3^{x+2} \cdot 4^y - 3^x \cdot 4^{y+1} = 11,520,… — GRE General Test

If $x$ and $y$ are positive integers such that $3^{x+2} \cdot 4^y - 3^x \cdot 4^{y+1} = 11,520$ , what is the value of $x + y$ ?

Factoring

3^x \cdot 4^y

from the expression

3^{x+2} \cdot 4^y - 3^x \cdot 4^{y+1}

yields

3^x \cdot 4^y (3^2 - 4^1) = 5 \cdot 3^x \cdot 4^y

. Setting this equal to 11,520 and dividing by 5 gives

3^x \cdot 4^y = 2,304

. Prime factorization of 2,304 gives

3^2 \cdot 4^4

, so

x = 2

and

y = 4

. The sum

x + y

is equal to 6.

Factor out the greatest common exponential factor

3^x \cdot 4^y

from the left side of the equation.

3^x \cdot 4^y (3^2 - 4^1) = 11,520

By exponent rules,

3^{x+2} = 3^x \cdot 3^2

and

4^{y+1} = 4^y \cdot 4^1

Evaluate the constant factor inside the parentheses.

3^2 - 4^1 = 9 - 4 = 5

, so

5 \cdot 3^x \cdot 4^y = 11,520

Simplifying numerical exponents.

Divide both sides of the equation by 5.

3^x \cdot 4^y = 2,304

Isolating the variable exponential terms.

Determine the prime factorization of 2,304 into powers of 3 and 4.

2,304 = 9 \cdot 256 = 3^2 \cdot 4^4

, which implies

x = 2

and

y = 4

Unique factorization for integer bases.

Calculate the sum

x + y

x + y = 2 + 4 = 6

Answering the explicit prompt.

Factoring Exponential Expressions and Unique Factorization

Question