1. **State the problem:** We need to find the value of $y$ when $x=13$ for the function $$y=\frac{3}{2}\sqrt{x+3}.$$\n\n2. **Formula and explanation:** The function involves a square root and a fraction multiplier. The square root function is defined for $x+3 \geq 0$, which is true for $x=13$.\n\n3. **Substitute $x=13$ into the function:**\n$$y=\frac{3}{2}\sqrt{13+3}$$\n\n4. **Simplify inside the square root:**\n$$y=\frac{3}{2}\sqrt{16}$$\n\n5. **Evaluate the square root:**\n$$y=\frac{3}{2} \times 4$$\n\n6. **Multiply:**\n$$y=\frac{3 \times 4}{2}$$\n\n7. **Simplify the fraction by canceling common factors:**\n$$y=\frac{\cancel{3} \times 4}{\cancel{2}} = 3 \times 2 = 6$$\n\n**Final answer:**\n$$y=6$$