jmc

To begin, we can combine all of these square roots into a single square root: $$\frac{\sqrt{160}}{\sqrt{252}}\times \frac{\sqrt{245}}{\sqrt{108}}=\sqrt{\frac{160}{252}}\times \sqrt{\frac{245}{108}}=\sqrt{\frac{160\cdot 245}{252\cdot 108}}$$Now, simplify under the square root by canceling common factors. To start, 160 and 108 are both divisible by 4. 252 and 160 also share a factor of 4. This leaves us with: $$\sqrt{\frac{10\cdot 245}{63\cdot 27}}$$Looking carefully, we can see that 63 and 245 both share a factor of 7. Cancel this, and simplify: $$\sqrt{\frac{10\cdot 35}{9\cdot 27}}=\frac{5}{9}\sqrt{\frac{14}{3}}=\boxed{\frac{5\sqrt{42}}{27}}$$