# 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 $$(2,3)$$.