#
Problem
of the Week

Problem
C and Solution

Exponential
Expressions

## Problem

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

**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 \((2,3)\).