Solution:
Given expressing,
$$ = \frac{1}{{\sqrt 1 + \sqrt 2 }}{\text{ + }}\frac{1}{{\sqrt 2 + \sqrt 3 }}$$     $$ + \frac{1}{{\sqrt 3 + \sqrt 4 }}$$   $$ + ...... + $$   $$\frac{1}{{\sqrt {120} + \sqrt {121} }}$$
$$ = \frac{1}{{\sqrt 2 + \sqrt 1 }}{\text{ + }}\frac{1}{{\sqrt 3 + \sqrt 2 }}$$     $$ + \frac{1}{{\sqrt 4 + \sqrt 3 }}$$   $$ + ...... + $$   $$\frac{1}{{\sqrt {121} + \sqrt {120} }}$$
$$ = \frac{1}{{\sqrt 2 + \sqrt 1 }} \times $$   $$\frac{{\sqrt 2 - \sqrt 1 }}{{\sqrt 2 - \sqrt 1 }}{\text{ + }}$$   $$\frac{1}{{\sqrt 3 + \sqrt 2 }} \times $$   $$\frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} + $$   $$\frac{1}{{\sqrt 4 + \sqrt 3 }} \times $$   $$\frac{{\sqrt 4 - \sqrt 3 }}{{\sqrt 4 - \sqrt 3 }} + $$   $$...... + $$   $$\frac{1}{{\sqrt {121} + \sqrt {120} }} \times $$   $$\frac{{\sqrt {121} - \sqrt {120} }}{{\sqrt {121} - \sqrt {120} }}$$
$$ = \frac{{\sqrt 2 - \sqrt 1 }}{{2 - 1}} + \frac{{\sqrt 3 - \sqrt 2 }}{{3 - 2}}$$     $$ + \frac{{\sqrt 4 - \sqrt 3 }}{{4 - 3}}$$   $$ + ...... + $$   $$\frac{{\sqrt {121} - \sqrt {120} }}{{121 - 120}}$$
$$ = \sqrt 2 - \sqrt 1 + \sqrt 3 - \sqrt 2 $$     $$ + \sqrt 4 - \sqrt 3 $$   $$ + ...... + $$   $$\sqrt {121} - \sqrt {120} $$
$$ = - 1 + \sqrt {121} $$
$$ = - 1 + 11$$
$$ = 10$$

Solution:
$$\eqalign{ = 10y\sqrt {{y^3} - {y^2}} \cr = 10 \times 5\sqrt {{5^3} - {5^2}} \cr = 50 \times \sqrt {125 - 25} \cr = 50 \times \sqrt {100} \cr = 50 \times 10 \cr = 500 \cr} $$

Solution:
$$3600 = {2^3} \times {5^2} \times {3^2} \times 2$$
To make it a perfect cube, it must be divided by   $${5^2} \times {3^2} \times 2,i.e.,450$$

Solution:
$$\eqalign{ \,\,\,\,\,\,\,1|\overline 1 \,.\,\overline {56} \,\,\overline {25} \,(1.25 \cr \,\,\,\,\,\,\,\,\,\,|1 \cr \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr \,\,\,22|\,\,\,\,\,\,56 \cr \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,44 \cr \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr 245\,|\,\,\,\,\,\,\,12\,25 \cr \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,12\,25 \cr \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\text{x} \cr \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr \therefore \sqrt {1.5625} = 1.25 \cr} $$

Solution:
$$\eqalign{ = \sqrt {\frac{{0.16}}{{0.4}}} \cr = \sqrt {\frac{{0.16}}{{0.40}}} \cr = \sqrt {\frac{{16}}{{40}}} \cr = \sqrt {\frac{4}{{10}}} \cr = \sqrt {0.4} \cr = 0.63 \cr} $$

Solution:
$$\eqalign{ {\text{Given expression,}} \cr = \sqrt {176 + 49} \cr = \sqrt {225} \cr = 15 \cr} $$

Solution:
Sum of prime numbers greater than 4 but less than 16
$$\eqalign{ = \left( {5 + 7 + 11 + 13} \right) \cr = 36 \cr \therefore \frac{1}{4} \times 36 \cr = 9 \cr = {3^2} \cr} $$