Solution:

\begin{align*} &= 3\sqrt{2} + \sqrt [4] {2500} + \sqrt [4] {64} +6 \sqrt{8}\\&= 3\sqrt{2}+ \sqrt[4]{5 \times 5 \times 5 \times 5 \times 2 \times 2}+ \sqrt [4]{2 \times 2 \times 2 \times 2\times 2 \times 2}+ 6\sqrt{2 \times 2 \times 2}\\&= 3\sqrt{2}+5\sqrt[4]{4}+2\sqrt[4]{4}+12\sqrt{2}\\&=15\sqrt{2}+7\sqrt[4]{4} :_\text {Ans.} \end{align*}

\begin{align*} \sqrt [3] {16} + \sqrt [3] {54} - \sqrt [3] {250} &= \sqrt [3] {2^3 \times 2} + \sqrt [3] {3^3 \times 2} - \sqrt [3] {5^3 \times 2}\\ &= 2\sqrt [3]{2} + 3\sqrt [3]2 - 5\sqrt [3]2\\ &= 5\sqrt [3]2 - 5\sqrt [3]3\\ &= 0_{Ans}\end{align*}

\begin{align*} \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}} \times \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 {a + b + a - b - 2 \sqrt {a + b}\sqrt {a - b}}{a + b - a + b}\\ &= \frac {2a - 2\sqrt {a^2 - b^2}}{2b}\\ &= \frac {2(a - \sqrt {a^2 - b^2})}{2b}\\ &= \frac {a - \sqrt {a^2 - b^2}}b_{Ans}\\ \end{align*}

\(\frac {\sqrt y + \sqrt 5}{\sqrt y - \sqrt 5}\) = 3

or, \(\sqrt y\) + \(\sqrt 5\) = 3\(\sqrt y\) - 3\(\sqrt 5\)

or, 3\(\sqrt y\) - \(\sqrt y\) = \(\sqrt 5\) + 3\(\sqrt 5\)

or, 2\(\sqrt y\) = 4\(\sqrt 5\)

or, \(\sqrt y\) = \(\frac {4\sqrt 5}{2}\)

or, \(\sqrt y\) = 2\(\sqrt 5\)

Squaring on both sides,

(\(\sqrt y\))²= (2\(\sqrt 5\))²

or, y = 4× 5

∴ y = 20_Ans

\(\frac {2y + 3}{\sqrt y - 1}\) = \(\frac 13\) (\(\sqrt y\) + 1)

or, 6y + 9 = (\(\sqrt y\) + 1) (\(\sqrt y\) - 1)

or, 6y + 9 = (\(\sqrt y\))² - (1)²

or, 6y + 9 = y - 1

or, 6y - y = -1 -9

or, 5y = - 10

or, y = \(\frac {-10}5\)

∴ y = -2_Ans

\(\sqrt x\) + 1 = 5 - \(\frac {\sqrt x -1}2\)

or, \(\sqrt x\) + 1 - 5 = \(\frac {- (\sqrt x - 1)}{2}\)

or, \(\sqrt x\) - 4 = \(\frac {-\sqrt x + 1}{2}\)

or, 2\(\sqrt x\) - 8 + \(\sqrt x\) = 1

or, 3\(\sqrt x\) = 1 + 8

or, 3\(\sqrt x\) = 9

or, \(\sqrt x\) = \(\frac 93\)

or, \(\sqrt x\) = 3

Squaring on both sides,

(\(\sqrt x\))² = 3²

∴ x = 9 _Ans

\(\sqrt x\) + \(\sqrt {x - 20}\) = 10

or, \(\sqrt {x - 20}\) = 10 - \(\sqrt x\)

Squaring on both sides,

(\(\sqrt {x - 20}\))² = (10 - \(\sqrt x\))²

or, x - 20 = 100 - 20\(\sqrt x\) + x

or, 20\(\sqrt x\) = 100 + 20

or, \(\sqrt x\) = \(\frac {120}{20}\)

or, \(\sqrt x\) = 6

Squaring on both sides,

(\(\sqrt x\))² = 6²

∴ x = 36_Ans

\(\sqrt {4 (x + 1)}\) = \(\sqrt {4x}\) + 1

Squaring on both sides,

(\(\sqrt {4 (x + 1)}\))² = (\(\sqrt {4x}\) + 1)²

or, 4(x + 1) = 4x + 2\(\sqrt {4x}\) + 1

or, 4x + 4 - 4x - 1 = 2\(\sqrt {4x}\)

or, \(\frac 32\) = \(\sqrt {4x}\)

Again,

Squaring on both sides,

(\(\sqrt {4x}\))² = (\(\frac 32\))²

or, 4x = \(\frac 94\)

or, x = \(\frac 9{4 \times 4}\)

∴ x = \(\frac 9{16}_{Ans}\)