Solution:

= $$\frac{\sqrt{a+b} + \sqrt{a−b}}{\sqrt{a+b} − \sqrt{a−b}}$$ × $$\frac{\sqrt{a+b} + \sqrt{a−b}}{\sqrt{a+b} + \sqrt{a−b}}$$
= $$\frac{(\sqrt{a+b} + \sqrt{a−b})^2}{(\sqrt{(a+b)^2} − \sqrt{(a−b)^2}}$$
= $$\frac{(\sqrt{a+b})^2 + 2 . \sqrt{a+b} . \sqrt{a−b} + (\sqrt{a−b})^2}{a + b − a + b}$$
= $$\frac{a + b + 2\sqrt{a^2 − b^2} + a − b}{2b}$$
= $$\frac{2a + 2\sqrt{a^2 − b^2}}{2b}$$
= $$\frac{2 (a + \sqrt{a^2 − b^2})}{2}$$
= a + $$\sqrt{a^2 − b^2}$$

Solution:

=$$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5}− 3}$$ + $$\frac{\sqrt{5} − \sqrt{3}}{\sqrt{5} + \sqrt{3}}$$
= $$\frac{(\sqrt{5} + \sqrt{3})^2 + (\sqrt{5} − \sqrt{3})^2}{(\sqrt{5})^2 − (\sqrt{3})^2}$$
= $$\frac{(\sqrt{5})^2 + 2 . \sqrt{5} . \sqrt{3} + (\sqrt{3})^2 + (\sqrt{5})^2− 2 . \sqrt{5} . \sqrt{3} + (\sqrt{3})^2}{5− 3}$$
= $$\frac{5 + 2 \sqrt{15} + \sqrt{3} + \sqrt{5} − 2 \sqrt{15} + \sqrt{3}}{2}$$
= $$\frac{5 + 3 + 5 + 3}{2}$$
= $$\frac{16}{2}$$
= 8

Solution:

Let 3$$\sqrt{5}$$ = a
2$$\sqrt{3}$$ = b
Now,
= $$\frac{3}{a+b}$$
= $$\frac{3}{a+b}$$× $$\frac{a−b}{a−b}$$
= $$\frac{3(3\sqrt{5}− 2\sqrt{3}}{(3\sqrt{5})^2 − (2\sqrt{3})^2}$$
= $$\frac{9\sqrt{5}− 6\sqrt{3}}{45− 12}$$
= $$\frac{3(3\sqrt{5}− 2\sqrt{3})}{33}$$
= $$\frac{3\sqrt{5}− 2\sqrt{3}}{11}$$

Solution:

= $$\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} − \sqrt{2}}$$
=$$\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} − \sqrt{2}}$$ × $$\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$$
= $$\frac{(\sqrt{5} + \sqrt{2}}{(\sqrt{5})^2 − (\sqrt{2})^2}$$
= $$\frac{(\sqrt{5})^2 + 2 . \sqrt{5} . \sqrt{2} + (\sqrt{2})^2}{5 − 2}$$
= $$\frac{5 + 2\sqrt{10} + 2}{3}$$
= $$\frac{7 + 2\sqrt{10}}{3}$$

Solution:

=$$\sqrt[3]{54}$$ × $$\sqrt[3]{625}$$
= $$\sqrt[3]{54 × 625}$$
= $$\sqrt[3]{8 × 3 × 125 × 5}$$
= $$\sqrt[3]{2^3 × 5^3 × 15}$$
= 2 × 5$$\sqrt[3]{15}$$
= 10$$\sqrt[3]{15}$$

Solution:

= 2$$\sqrt{24} + 7\sqrt{54}$$
= 2$$\sqrt{4×6}$$ + 7$$\sqrt{9×6}$$
= 2$$\sqrt{2^2× 6}$$ + 7$$\sqrt{3^3 × 6}$$
= 2 × 2$$\sqrt{6}$$ + 7 × 3$$\sqrt{6}$$
= 4$$\sqrt{6}$$ + 21$$\sqrt{6}$$
= 4 + 21$$\sqrt{6}$$
= 25$$\sqrt{6}$$

Solution:

= 8$$\sqrt{25 × 3}$$ − 2$$\sqrt{16 × 3}$$
= 8$$\sqrt{5^2 × 3}$$ − 2$$\sqrt{4^2 × 3}$$
= 8 × 5$$\sqrt{3}$$ − 2 × 4$$\sqrt{3}$$
= 40$$\sqrt{3}$$ − 8$$\sqrt{3}$$
= 40 − 8$$\sqrt{3}$$
= 32$$\sqrt{3}$$

Solution:

= 10$$\sqrt[3]{81}$$ − 12$$\sqrt[3]{24}$$
= 10$$\sqrt[3]{3^3×3}$$ − 12$$\sqrt[3]{2^3 × 3}$$
= 10 × 3$$\sqrt[3]{3}$$ − 12 × 2$$\sqrt[3]{3}$$
= 30$$\sqrt[3]{3}$$ − 24$$\sqrt[3]{3}$$
= 30 − 24$$\sqrt[3]{3}$$
= 6$$\sqrt[3]{3}$$

Solution:

= $$\sqrt{27}$$ − 4$$\sqrt{3}$$ + 3$$\sqrt{12}$$
= $$\sqrt{3^2 × 3}$$ − 4$$\sqrt{3}$$ + 3$$\sqrt{2^2 × 3}$$
= 3$$\sqrt{3}$$ − 4$$\sqrt{3}$$ + 3 × 2$$\sqrt{3}$$
= 3 − 4 + 6$$\sqrt{3}$$
= 5$$\sqrt{3}$$

Solution:

= $$\sqrt{20}$$ + 2$$\sqrt{7}$$ + 4$$\sqrt{45}$$ − $$\sqrt{28}$$
= $$\sqrt{2^2 × 5}$$ + 2$$\sqrt{7}$$ + 4$$\sqrt{3^2 × 5}$$− $$\sqrt{2^2 × 7}$$
= 2$$\sqrt{5}$$ + 2$$\sqrt{7}$$ + 4 × 3$$\sqrt{5}$$ − 2$$\sqrt{7}$$
= 2$$\sqrt{5}$$ + 12$$\sqrt{5}$$
= 2 + 12$$\sqrt{5}$$
= 14$$\sqrt{5}$$

Solution:

= $$\frac{1}{\sqrt{2} + 1}$$
= $$\frac{1}{\sqrt{2} + 1}$$ × $$\frac{\sqrt{2} − 1}{\sqrt{2} − 1}$$
= $$\frac{\sqrt{2} − 1}{(\sqrt{2})^2 − (\sqrt{1})^2}$$
= $$\frac{\sqrt{2} − 1}{2 − 1}$$
= $$\sqrt{2}$$ − 1