To solve the equation √x+3 -√16x+48 =0, we can follow these steps:

1. Move the term √16x+48 to the other side of the equation:
√x+3 = √16x+48

2. Square both sides of the equation to eliminate the square root:
(√x+3)^2 = (√16x+48)^2

3. Simplify both sides of the equation:
x + 6√x + 9 = 16x + 48

4. Move the terms involving x to one side of the equation and the constant terms to the other side:
x - 16x = 48 - 9 - 6√x

-15x = 39 - 6√x

5. Divide both sides of the equation by -15:
x = (39 - 6√x) / -15

6. Multiply both sides of the equation by -15 to eliminate the fraction:
-15x = 39 - 6√x

7. Move the term -6√x to the other side of the equation:
-15x + 6√x = 39

8. Factor out the common factor of √x:
√x(-15 + 6) = 39

9. Simplify the equation:
-9√x = 39

10. Divide both sides of the equation by -9:
√x = -39/9

11. Square both sides of the equation to eliminate the square root:
x = (-39/9)^2

12. Simplify the equation:
x = 1521/81

Therefore, the solution to the equation √x+3 -√16x+48 =0 is x = 1521/81.