[1/2]^x + 1/3]^x+3 = 9

To solve the equation [1/2]^x + [1/3]^x + 3 = 9, we can start by simplifying the equation.

Let's rewrite the equation as follows:

(1/2)^x + (1/3)^x + 3 = 9

Now, let's subtract 3 from both sides:

(1/2)^x + (1/3)^x = 6

Next, let's find a common denominator for the fractions:

(3/6)^x + (2/6)^x = 6

Now, let's combine the fractions:

(3^x + 2^x) / 6^x = 6

To simplify further, let's multiply both sides by 6^x:

3^x + 2^x = 6 * 6^x

Now, let's rewrite 6 as 2 * 3:

3^x + 2^x = 2 * 3 * 6^x

Since 6^x = (2 * 3)^x = 2^x * 3^x, we can substitute this into the equation:

3^x + 2^x = 2 * 3 * (2^x * 3^x)

Simplifying further:

3^x + 2^x = 2^x * 3^(x+1)

Now, let's divide both sides by 2^x:

(3^x + 2^x) / 2^x = 3^(x+1)

Since 2^x / 2^x = 1, we have:

3^x / 2^x + 1 = 3^(x+1)

Now, let's subtract 1 from both sides:

3^x / 2^x = 3^(x+1) - 1

To simplify further, let's rewrite 3^(x+1) as 3 * 3^x:

3^x / 2^x = 3 * 3^x - 1

Now, let's multiply both sides by 2^x:

3^x = 2^x * (3 * 3^x - 1)

Expanding the equation:

3^x = 2^x * 3 * 3^x - 2^x

Now, let's subtract 2^x * 3 * 3^x from both sides:

3^x - 2^x * 3 * 3^x = -2^x

Factoring out 3^x:

3^x(1 - 2^x * 3) = -2^x

Now, let's divide both sides by (1 - 2^x * 3):

3^x = -2^x / (1 - 2^x * 3)

At this point, it seems difficult to find an exact solution for x. However, we can use numerical methods or approximation techniques to find an approximate solution.