If x and y are positive integers such that 2^{x+1} + 2^x = 3^{y+2} - 3^y, what is the valu… — GRE General Test

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

Answer: 4

The value of

x + y

4

Factoring the left side gives

2^x(2 + 1) = 3 \cdot 2^x

, while factoring the right side gives

3^y(9 - 1) = 8 \cdot 3^y = 2^3 \cdot 3^y

. Equating the two expressions gives

3 \cdot 2^x = 2^3 \cdot 3^y

. Rearranging terms to separate bases yields

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

. Because 2 and 3 share no common prime factors, this equality holds for integers if and only if both exponents are equal to 0. Solving

x - 3 = 0

gives

x = 3

, and solving

y - 1 = 0

gives

y = 1

. Both are positive integers. Thus,

x + y = 3 + 1 = 4

Factor out common terms on both sides of the equation.

2^x(2 + 1) = 3^y(9 - 1) \implies 3 \cdot 2^x = 8 \cdot 3^y

Factoring simplifies sums of powers with identical bases.

Rewrite integers using prime factorizations and re-group bases.

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

Dividing both sides by

2^3 \cdot 3

separates the base-2 and base-3 exponential terms.

Set each exponent to zero using prime independence.

x - 3 = 0 \implies x = 3

and

y - 1 = 0 \implies y = 1

Powers of distinct prime numbers 2 and 3 can only be equal if both powers equal

1

(

2^0 = 3^0 = 1

Calculate the required sum

x + y

3 + 1 = 4

Evaluates the requested combined value of the variables.

Solving exponential equations involving distinct prime bases through factoring and exponent properties.

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