given 2^m+1 + 2^m = 3 ^n+2 - 3^n where m and n are integers.
determine the value of m+n.
To solve this equation, let's first simplify both sides by factoring out a common term:
2^m+1 + 2^m = 3^n+2 - 3^n
Rewrite 2^m+1 as 2 * 2^m and rewrite 3^n+2 as 3^2 * 3^n:
2 * 2^m + 2^m = 3^2 * 3^n - 3^n
Next, combine the like terms on both sides:
3 * 2^m = (3^2 - 1) * 3^n
Simplify the right side of the equation:
3 * 2^m = 8 * 3^n
Divide both sides of the equation by 3:
2^m = 8 * 3^(n-1)
8 is equal to 2^3, so we can rewrite the right side of the equation:
2^m = 2^3 * 3^(n-1)
To have two sides of the equation equal, the exponents of 2 and 3 must be equal:
m = 3 + (n - 1)
m = n + 2
Now, we have two equations:
2^m = 2^3 * 3^(n-1)
m = n + 2
Since the left side of the first equation is equal to 2^m, and the right side is equal to 2^3 * 3^(n-1), we can equate the exponents:
m = 3 + (n - 1)
m = n + 2
By comparing the two equations, we can conclude that m = 3 and n = 1.
Therefore, the value of m + n is 3 + 1 = 4.