91Ó°ÊÓ

First, we simplify the radicands by finding the prime factorization of the radicands and then grouping the common factors. For the first term: $$6\sqrt{48a^4}$$ Breaking down 48 into its prime factors: $$2^4 * 3$$ Breaking down $$a^4$$ into its prime factors: $$a^2 * a^2$$ For the second term: $$2 a\sqrt{27a^2}$$ Breaking down 27 into its prime factors: $$3^3$$ Breaking down $$a^2$$ into its prime factors: $$a^2$$ For the third term: $$-3a^2\sqrt{75}$$ Breaking down 75 into its prime factors: $$3 * 5^2$$ Now we can rewrite the terms with simplified radicands. First Term: $$6\sqrt{(2^4 * 3)(a^2 * a^2)} = 6\sqrt{(2^2 * 2^2 * 3)(a^2 * a^2)} = 6(2^2 * a^2)\sqrt{2^2 * 3} = 4 a^2\sqrt{12}$$ Second Term: $$2 a\sqrt{(3^3)(a^2)} = 2 a(a\sqrt{3^3}) = 2a^2\sqrt{27}$$ Third Term: $$-3a^2\sqrt{75} = -3a^2\sqrt{(3 * 5^2)} = -3a^2(5\sqrt{3}) = -15 a^2 \sqrt{3}$$ Now our simplified equation is: $$4 a^2\sqrt{12} + 2a^2\sqrt{27} -15 a^2 \sqrt{3}$$

Now that we have simplified the radicals, check to see if there are any like terms that can be combined. In this case, none of the radicals are exactly alike, so we cannot combine any further. Thus, the final simplified expression is: $$4 a^2\sqrt{12} + 2a^2\sqrt{27} -15 a^2 \sqrt{3}$$