Simplify the following expression and find the value of x:

Step-by-Step Solution

To find the value of x, we need to simplify the given exponential expressions by expressing both sides with the same base.

1. Simplify the equations:

   • Given equation 1: 3^(x - y) = 27

     • We know that 27 = 3^3.

     • Therefore: 3^(x - y) = 3^3

     • Equating the exponents: x - y = 3 --- (Equation 1)

   • Given equation 2: 3^(x + y) = 243

     • We know that 243 = 3^5.

     • Therefore: 3^(x + y) = 3^5

     • Equating the exponents: x + y = 5 --- (Equation 2)

2. Solve the system of linear equations:

   • Now we have a simple system of two linear equations:

     1. x - y = 3

     2. x + y = 5

   • Add the two equations together: (x - y) + (x + y) = 3 + 5 2x = 8

   • Solve for x: x = 4