If x and y are positive integers such that 3^x \cdot 5^y = 675, what is the value of 2^{x+… — GMAT

If $x$ and $y$ are positive integers such that $3^x \cdot 5^y = 675$ , what is the value of $2^{x+y} \cdot 3^{x-y}$ ?

Prime factorizing

675

gives

3^3 \cdot 5^2

. Equating the exponents yields

x = 3

and

y = 2

. Substituting these into

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

yields

2^5 \cdot 3^1 = 32 \cdot 3 = 96

Find the prime factorization of 675.

675 = 27 \times 25 = 3^3 \cdot 5^2

Decomposing 675 into prime factors allows matching the exponents of prime bases 3 and 5.

Equate exponents of matching prime bases to determine

x

and

y

x = 3

and

y = 2

Since 3 and 5 are prime numbers, the prime factorization of 675 is unique.

Evaluate the target expression

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

2^{3+2} \cdot 3^{3-2} = 2^5 \cdot 3^1 = 32 \cdot 3 = 96

Substitute the determined values of

x

and

y

into the target power expression.

Prime Factorization and Exponent Properties

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