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