I will assume the square root extends all the way across.
\(\sqrt{48a^4b^5c^3}\)
\(\sqrt{2^4\cdot3\cdot a^4b^5c^3}\)
\(\sqrt{(2^2)^2\cdot3\cdot(a^2)^2\cdot(b^2)^2\cdot b\cdot c^2\cdot c}\)
\(\sqrt{(4)^2\cdot3\cdot(a^2)^2\cdot(b^2)^2\cdot b\cdot c^2\cdot c}\)
\(\sqrt{(4\cdot a^2 \cdot b^2 \cdot c)^2\cdot3\cdot b\cdot c}\)
\(4\cdot a^2\cdot b^2\cdot c\sqrt{3\cdot b\cdot c}\)
\(\boxed{4a^2b^2c\sqrt{3bc}}\)
