We are given two expressions:
Expression \(A\): \(72\times 7^x\)
Expression \(B\): \(441\times 2^y\)
Given that \(x\) and \(y\) are positive integers, find all ordered pairs \((x,y)\) so that the value of Expression \(A\) is equal to the value of Expression \(B\).
Solution 1
We write each expression as the product of prime numbers.
Expression \(A=(2^3)(3^2)(7^x)\) and Expression \(B=(3^2)(7^2)(2^y).\)
Since \(x\) and \(y\) are each positive integers and the expressions are equal in value, then the corresponding exponents for each prime number must be equal. Therefore, \(x=2\) and \(y=3\) is the only integer solution.
Thus, the only ordered pair is \((2,3)\).
Solution 2
Setting the two expressions equal to each other, we have \[72 \times 7^x=441 \times 2^y\] Dividing both sides by \(9\), we have \[8\times 7^x=49 \times 2^y\] Expressing each side of the equation as the product of prime numbers, we have \[2^3 \times 7^x=7^2 \times 2^y\]
Since \(x\) and \(y\) are each positive integers and the expressions are equal in value, then the corresponding exponents for each prime number must be equal. Therefore, \(x=2\) and \(y=3\) is the only integer solution.
Thus, the only ordered pair is \((2,3)\).