Multiply and simplify products of radicals that have numerical coefficients and variable radicands.

ⓐ $(10\sqrt{6p^3})(4\sqrt{3p})$: Multiply the coefficients together and the radicands together: $(10\sqrt{6p^3})(4\sqrt{3p}) = 40\sqrt{18p^4}$ Simplify the resulting radical by extracting the largest perfect square factor ($9p^4$): $40\sqrt{18p^4} = 40\sqrt{9p^4} \cdot \sqrt{2}$ Simplify the perfect square and multiply it with the outside coefficient: $40 \cdot 3p^2 \cdot \sqrt{2} = 120p^2\sqrt{2}$

ⓑ $(2\sqrt[4]{20y^2})(3\sqrt[4]{28y^3})$: Multiply the coefficients and the radicands: $(2\sqrt[4]{20y^2})(3\sqrt[4]{28y^3}) = 6\sqrt[4]{560y^5}$ When the radicands involve large numbers, it is often advantageous to factor them to find perfect powers ($560 = 16 \cdot 35$). Identify the largest perfect fourth power factors: $6\sqrt[4]{16y^4 \cdot 35y} = 6\sqrt[4]{16y^4} \cdot \sqrt[4]{35y}$ Simplify the perfect fourth power and multiply: $6 \cdot 2y\sqrt[4]{35y} = 12y\sqrt[4]{35y}$