1. **State the problem:** Multiply $5\sqrt{3}$ by $6\sqrt{3}$. 2. **Recall the multiplication rule for radicals:** $a\sqrt{b} \times c\sqrt{d} = (a \times c) \times \sqrt{b \times d}$. 3. **Apply the rule:** $$5\sqrt{3} \times 6\sqrt{3} = (5 \times 6) \times \sqrt{3 \times 3}$$ 4. **Calculate the product of the coefficients:** $$5 \times 6 = 30$$ 5. **Calculate the product inside the square root:** $$\sqrt{3 \times 3} = \sqrt{9}$$ 6. **Simplify the square root:** $$\sqrt{9} = 3$$ 7. **Multiply the coefficient by the simplified radical:** $$30 \times 3 = 90$$ **Final answer:** $$5\sqrt{3} \times 6\sqrt{3} = 90$$