To solve the equation, we need to simplify the expression first.

Starting with the left side of the equation:
căn(4x-24) + 9 căn(x-6/9) - 2/3 căm(9x-54)

We can simplify each term separately:
căn(4x-24) = căn(4(x-6)) = 2 căn(x-6)
căn(x-6/9) = căn((x-6)/9)
căm(9x-54) = căm(9(x-6)) = 3 căm(x-6)

Now, the equation becomes:
2 căn(x-6) + 9 căn((x-6)/9) - 2/3 * 3 căm(x-6) = 4

Next, let's simplify the terms further:
2 căn(x-6) + 9 căn((x-6)/9) - 2 căm(x-6) = 4

Now, let's combine like terms:
2 căn(x-6) - 2 căm(x-6) + 9 căn((x-6)/9) = 4

Since we have both căn(x-6) and căm(x-6), we can simplify further:
2 căn(x-6) - 2 căm(x-6) = 2(căn(x-6) - căm(x-6))

Now, the equation becomes:
2(căn(x-6) - căm(x-6)) + 9 căn((x-6)/9) = 4

To solve for x, we need to isolate the square root terms:
2(căn(x-6) - căm(x-6)) = 4 - 9 căn((x-6)/9)

Dividing both sides by 2:
căn(x-6) - căm(x-6) = 2 - 9 căn((x-6)/9)

Now, we can square both sides to eliminate the square root terms:
(căn(x-6) - căm(x-6))^2 = (2 - 9 căn((x-6)/9))^2

Expanding both sides:
(x-6) - 2căn(x-6)căm(x-6) + (x-6) = 4 - 36 căn((x-6)/9) + 81(x-6)/9

Simplifying:
2(x-6) - 2căn(x-6)căm(x-6) = 4 - 36 căn((x-6)/9) + 9(x-6)

Combining like terms:
2x - 12 - 2căn(x-6)căm(x-6) = 4 - 36 căn((x-6)/9) + 9x - 54

Moving all terms to one side of the equation:
2x - 12 - 4 + 36 căn((x-6)/9) - 9x + 54 + 2căn(x-6)căm(x-6) = 0

Simplifying further:
-7x + 74 + 36 căn((x-6)/9) + 2căn(x-6)căm(x-6) = 0

This is the simplified form of the equation. To solve for x, we can use numerical methods or approximation techniques.