Answer
$\{3\}$
Work Step by Step
Step 1. Given the equation $\sqrt {3\sqrt {2x+3}}=\sqrt {5x-6}$
Step 2. Take square on both sides to get: $3\sqrt {2x+3}=5x-6$
Step 3. Take square on both sides again to get: $18x+27=25x^2-60x+36$ or $25x^2-78x+9=0$ which gives $x=\frac{3}{25},3$
Step 4. Check answers in the original equation: only $\{3\}$ fits.