If x and y are positive integers such that 2^x - 2^{x-2} = 3^y, what is the value of 2^{x+… — GMAT

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

The value of

2^{x+y} + 3^{x-y}

is 11.

Factoring the left side of the equation yields

2^{x-2}(2^2 - 1) = 3 \cdot 2^{x-2}

. Setting this equal to

3^y

and dividing both sides by 3 gives

2^{x-2} = 3^{y-1}

. Because 2 and 3 are coprime bases, equality is possible only when both exponents equal zero. This determines

x = 2

and

y = 1

, both of which are positive integers. Substituting these values into

2^{x+y} + 3^{x-y}

results in

2^3 + 3^1 = 8 + 3 = 11

Factor out the common term on the left side of the given equation

2^x - 2^{x-2} = 2^{x-2}(2^2 - 1) = 2^{x-2}(4 - 1) = 3 \cdot 2^{x-2}

Factoring out the smallest power of 2 simplifies the subtraction into a multiplication.

Equate the factored expression to the right side and simplify

3 \cdot 2^{x-2} = 3^y \implies 2^{x-2} = \frac{3^y}{3} = 3^{y-1}

Dividing both sides by 3 isolates the prime bases 2 and 3 on opposite sides of the equation.

Apply fundamental properties of prime factorizations

x - 2 = 0 \implies x = 2

and

y - 1 = 0 \implies y = 1

Since 2 and 3 are distinct prime numbers,

2^a = 3^b

can hold for integer exponents only if

a = 0

and

b = 0

Evaluate the target expression using

x = 2

and

y = 1

2^{2+1} + 3^{2-1} = 2^3 + 3^1 = 8 + 3 = 11

Substituting the unique positive integer solution pair

(2, 1)

into the expression yields the final value.

Uniqueness of Prime Base Exponent Equality

Question