Simplify the following expressions involving square roots: a) (2\[[2]5+3)([2]5+2) b) ([2]6+2)([2]3+[2]2) c) (4[2]3+[2]6)(4[2]3-[2]6)

$$(2\sqrt{5})(\sqrt{5}) + (2\sqrt{5})(2) + (3)(\sqrt{5}) + (3)(2)$$ $$2(5) + 4\sqrt{5} + 3\sqrt{5} + 6$$

$$10 + 7\sqrt{5} + 6$$ $$16 + 7\sqrt{5}$$

The simplified expression is $\boxed{16 + 7\sqrt{5}}$.

$$(\sqrt{6})(\sqrt{3}) + (\sqrt{6})(\sqrt{2}) + (2)(\sqrt{3}) + (2)(\sqrt{2})$$ $$\sqrt{18} + \sqrt{12} + 2\sqrt{3} + 2\sqrt{2}$$

$$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$

$$3\sqrt{2} + 2\sqrt{3} + 2\sqrt{3} + 2\sqrt{2}$$ $$(3\sqrt{2} + 2\sqrt{2}) + (2\sqrt{3} + 2\sqrt{3})$$ $$5\sqrt{2} + 4\sqrt{3}$$

The simplified expression is $\boxed{5\sqrt{2} + 4\sqrt{3}}$.

Here, $a = 4\sqrt{3}$ and $b = \sqrt{6}$.

$$(4\sqrt{3})^2 - (\sqrt{6})^2$$

$$(4^2 \times (\sqrt{3})^2) - 6$$ $$(16 \times 3) - 6$$ $$48 - 6$$ $$42$$

The simplified expression is $\boxed{42}$.

Here, $a = 3\sqrt{3}$ and $b = \sqrt{7}$.

$$(3\sqrt{3})^2 - 2(3\sqrt{3})(\sqrt{7}) + (\sqrt{7})^2$$

$$(3^2 \times (\sqrt{3})^2) - (2 \times 3 \times \sqrt{3 \times 7}) + 7$$ $$(9 \times 3) - 6\sqrt{21} + 7$$ $$27 - 6\sqrt{21} + 7$$

$$34 - 6\sqrt{21}$$

The simplified expression is $\boxed{34 - 6\sqrt{21}}$.

$$\frac{(4\sqrt{3})^2}{3^2} = \frac{4^2 \times (\sqrt{3})^2}{9}$$ $$\frac{16 \times 3}{9} = \frac{48}{9}$$

$$\frac{48}{9} = \frac{16}{3}$$

The simplified expression is $\boxed{\frac{16}{3}}$.