Given exp.   `sqrt(10+ sqrt(25 + sqrt( 108 + sqrt(154 + sqrt(225)))))`

                       `sqrt(10+ sqrt(25 + sqrt( 108 + sqrt(154 + 15    ))))`

                       =`sqrt(10 +sqrt(25 + sqrt(108 +sqrt(169))))`=

                       =`sqrt(10 + sqrt(25 + sqrt(108  + 13)))`       =  `sqrt(10 + sqrt(25 + sqrt(121)))`

                    = `sqrt (10 + sqrt(25 + 11))`                       =  `sqrt(10  +sqrt(36))`

                      `sqrt(10 + 6)`         =   `sqrt(16)`        = 4.

Answer : Option A

Explanation :

140√x + 315 = 1015
=> 140√x = 1015 - 315 = 700 = 140 × 5
=> √x = 5
=> x = 5² = 25

√ 50 x√ 98 = √ 50 x 98

=>√ 4900 
70.

Answer : Option C

Explanation :

$MF#%\left(\sqrt{7} - \dfrac{1}{\sqrt{7}}\right)^2 = \left(\sqrt{7}\right)^2 - 2 \times \sqrt{7} \times \dfrac{1}{\sqrt{7}} + \left(\dfrac{1}{\sqrt{7}}\right)^2 \\\\ = 7 - 2 + \dfrac{1}{7} = 5 + \dfrac{1}{7} = \dfrac{36}{7}$MF#%

Given exp. = √((9.5 * 0.0085 * 18.9) / (0.0017 * 1.9 * 0.021))
Now,
since the sum of decimal places in the numerator and denominator under
the radical sign is the same, we remove the decimal.
Given exp = √((95 * 85 * 18900) / (17 * 19 * 21))
= √(5 * 5 * 900)
= 5 * 30
= 150.

L.C.M. of 10, 12, 15, 18 = 180. Now, 180 = 2 * 2 * 3 * 3 *5 = 22 * 32 * 5.
To make it a perfect square, it must be multiplied by 5.
Required number = (22 * 32 * 52) = 900.

√(25 / 16)
= √(25 / 16)
= 5 / 4

Answer : Option D

Explanation :

$MF#%\dfrac{1}{(\sqrt{9}-\sqrt{8})} = \dfrac{(\sqrt{9}+\sqrt{8})}{(\sqrt{9}-\sqrt{8})(\sqrt{9}+\sqrt{8})}=\dfrac{(\sqrt{9}+\sqrt{8})}{9-8}= \sqrt{9}+\sqrt{8}\\\\$MF#%

$MF#%\text{Similarly all other terms can be rewritten. Thus,}\\\\
\dfrac{1}{(\sqrt{9}-\sqrt{8})}-\dfrac{1}{(\sqrt{8}-\sqrt{7})}+\dfrac{1}{(\sqrt{7}-\sqrt{6})}-\dfrac{1}{(\sqrt{6}-\sqrt{5})}+\dfrac{1}{(\sqrt{5}-\sqrt{4})}\\\\
= (\sqrt{9}+\sqrt{8}) - (\sqrt{8}+\sqrt{7}) + (\sqrt{7}+\sqrt{6}) - (\sqrt{6}+\sqrt{5}) + (\sqrt{5}+\sqrt{4})\\\\
=\sqrt{9}+\sqrt{4} = 3 + 2 = 5$MF#%

Clearly, a * b =
√ a² + b². 
Therefore, 5 * 12=√ 5² + 12².
 =>√ 25 + 144.
 =>√ 169.
 =>13.

Answer : Option A

Explanation :

$MF#%(0.000729)^{1/3} = \left(\dfrac{729}{10^6}\right)^{1/3} = \dfrac{9}{10^2} = \dfrac{9}{100} = .09$MF#%