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