Answer:
[tex]\large\boxed{\dfrac{4\sqrt3}{19}}[/tex]
Step-by-step explanation:
[tex]\text{Geometric mean of}\ x\ \text{and}\ y:\ \sqrt{xy}\\\\\text{We have}\ x=\dfrac{4\sqrt2}{19}\ \text{and}\ y=\dfrac{6\sqrt2}{19}.\\\\\text{Substitute:}\\\\\sqrt{\left(\dfrac{4\sqrt2}{19}\right)\left(\dfrac{6\sqrt2}{19}\right)}=\sqrt{\dfrac{24(\sqrt2)^2}{19^2}}=\sqrt{\dfrac{(24)(2)}{19^2}}=\dfrac{\sqrt{48}}{\sqrt{19^2}}=\dfrac{\sqrt{(16)(3)}}{19}\\\\=\dfrac{\sqrt{16}\cdot\sqrt3}{19}=\dfrac{4\sqrt3}{19}[/tex]
