\(\sqrt { 98 } =\sqrt { 49\times 2 } =7\sqrt { 2 } ;\sqrt { 18 } =\sqrt { 9\times 2 } 3\sqrt { 2 } \)
\(\sqrt { 50 } =\sqrt { 25\times 2 } =5\sqrt { 2 } ;\sqrt { 75 } =\sqrt { 25\times 3 } 5\sqrt { 3 } \)
\(\sqrt { 27 } =\sqrt { 9\times 3 } 3\sqrt { 3 } \)
∴ \(\sqrt { 98 } +\sqrt { 50 } -\sqrt { 18 } +\sqrt { 75 } -\sqrt { 27 } =7\sqrt { 2 } +5\sqrt { 2 } -3\sqrt { 2 } +5\sqrt { 3 } -3\sqrt { 3 } \)
= \(9\sqrt { 2 } +2\sqrt { 3 } \)

Given \(log_2\left({{\sqrt [ 3 ]{4 } }\over{4^2\sqrt{8}}} \right)\)
= \({log}_{2}\sqrt [ 3 ]{4 }-{log}_{2}4^2(\sqrt{8})\)
= \(log_24^{1/3}-[log_24^2+log_2\sqrt{8}]\)
= log²(2²)^1/3- log²(2²)²- log²(2³)^1/2
= log²2^1/3- log²2⁴- log²2^3/2
\(={{2}\over{3}}(1)-4(1)-{{3}\over{2}}(1)\)   \([\because {log}^{2}_{2}=1]\)
\(={{4-24-9}\over{6}}={{-29}\over{6}}\)

Given quadratic equation is \(\sqrt [ 8 ]{{{x}\over{x+3}} } -\sqrt{{{x+3}\over{x}}}=2.\)
Let y \(=\sqrt{x\over x+3}\Rightarrow{1\over y}=\sqrt{x+3\over x}\)
∴ (1) becomes 8y - \(\frac{1}{y}\) = 2

\(⇒\ {8y^2-1\over y}=2⇒8y^2-1=2\)
8y²-2y-1 = 0
(2y - 1)(4y + 1) = 0
2y = 1 or 4y = -1
\(y={1\over 2}\)or \(y={-1\over 4}\)
\(\sqrt{x\over x+3}={1\over 2}\sqrt{x\over x+3}={1\over 2}\ or\ \sqrt{x\over x+3}={-1\over 4}\)
Case(i) \(\sqrt{x\over x+3}={1\over 2}\Rightarrow{x\over x+3}={-1\over 4}\)
4x = x + 3 ⇒ 3x = 3 ⇒ x = 1
Case(ii) \(\sqrt{x\over x+3}={-1\over 4}\)
This is impossible since LHS is non-negative.
∴ The root is 1.

Given x^2/3 + x^1/3 - 2 = 0.
\(\Rightarrow\) (x^1/3)² + x^1/3-2 = 0
Let x^1/3= y
\(\Rightarrow\)y² + y - 2 = 0
\(\Rightarrow\)   \(y={{-1\pm\sqrt{{(1)}^{2}-4(1)(-2)}}\over{2}}\)
\(\left[{ \because y={{-b\pm\sqrt{{b}^{2}-4ac}}\over{2a}} a=1, \ \ \ b=1, \ \ \ c=-2}\right] \)                                      
\(\Rightarrow\)  \(y={{-1\pm\sqrt{9}}\over{}2}\)
\(\Rightarrow\)  \(y={{-1\pm3}\over{2}}\Rightarrow y=1,-2\Rightarrow x^{1/3}=1\) or -2.
Case (i) When x^1/3 = 1 \(\Rightarrow\) x^1/3 = 1^1/3\(\Rightarrow\) x = 1
Case (ii) When x^1/3 = - 2 \(\Rightarrow\) x = (-2)³ = -8
\(\therefore\) The roots are 1, -8.

Let the lengths of pieces of board x cm, y cm and z cm.
Let x < y < z.
Given x + y + z \(\le\) 91  ...(1)
y = x + 3 ..(2)
z = 2x  ...(3)
and Z \(\ge\) y + 5 ...(4)
\(\therefore\) (1) \(\Rightarrow\) x + (x +3) + 2x \(\le\) 91  
\(\Rightarrow\)4x+3\(\le\) 91
4x \(\le\) 88
x \(\le\)22
From (4), Z \(\ge\) x + 3 + 5 ..(5)
\(\Rightarrow\) 2x \(\ge \) x + 8
\(\Rightarrow\)x \(\ge\) 8  ..(6)
From (5) and (6), the length of the shortest board lies between 8 cm and 22 cm.
i.e x \(\in\) [ 8, 22 ]