$$\eqalign{ & {\text{Let the number be }}x \cr & {\text{Then,}} \cr & \Leftrightarrow \frac{3}{5}{x^2} = 126.15 \cr & \Leftrightarrow {x^2} = \left( {126.15 \times \frac{5}{3}} \right) \cr & \Leftrightarrow {x^2} = 210.25 \cr & \Leftrightarrow x = \sqrt {210.25} \cr & \Leftrightarrow x = 14.5 \cr} $$

$$\eqalign{ & \Leftrightarrow \frac{{52}}{x}{\text{ = }}\sqrt {\frac{{169}}{{289}}} \cr & \Leftrightarrow \frac{{52}}{x} = \frac{{13}}{{17}} \cr & \Leftrightarrow x = \left( {\frac{{52 \times 17}}{{13}}} \right) \cr & \Leftrightarrow x = 68 \cr} $$

$$\eqalign{ & \Leftrightarrow \sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} \cr & \Leftrightarrow x + \frac{x}{y} = {x^2}.\frac{x}{y} \cr & \Leftrightarrow \frac{{xy + x}}{y} = \frac{{{x^3}}}{y} \cr & \Leftrightarrow xy + x = {x^3} \cr & \Leftrightarrow y + 1 = {x^2} \cr & \Leftrightarrow y = {x^2} - 1 \cr} $$

$$\eqalign{ & = {\left( {\sqrt 3 - \frac{1}{{\sqrt 3 }}} \right)^2} \cr & = {\left( {\sqrt 3 } \right)^2} + {\left( {\frac{1}{{\sqrt 3 }}} \right)^2} - 2 \times \sqrt 3 \times \frac{1}{{\sqrt 3 }} \cr & = 3 + \frac{1}{3} - 2 \cr & = 1 + \frac{1}{3} \cr & = \frac{4}{3} \cr} $$

Given expression,
$$ = \sqrt {\frac{{11025}}{{100}}} \times \sqrt {\frac{1}{{100}}} \, \div \,$$    $$\sqrt {\frac{{25}}{{10000}}} \, - \,$$ $$\,\sqrt {\frac{{42025}}{{100}}} $$
$$\eqalign{ & = \frac{{105}}{{10}} \times \frac{1}{{10}} \div \frac{5}{{100}} - \frac{{205}}{{10}} \cr & = \frac{{105}}{{100}} \times \frac{{100}}{5} - \frac{{205}}{{10}} \cr & = 21 - \frac{{205}}{{10}} \cr & = \frac{{210 - 205}}{{10}} \cr & = \frac{5}{{10}} \cr & = \frac{1}{2} \cr & = 0.50 \cr} $$

$$\eqalign{ & \Rightarrow \sqrt {1 + \frac{x}{{169}}} = \frac{{14}}{{13}} \cr & \Rightarrow 1 + \frac{x}{{169}} = \frac{{196}}{{169}} \cr & \Rightarrow \frac{x}{{169}} = \left( {\frac{{196}}{{169}} - 1} \right) \cr & \Rightarrow \frac{x}{{169}} = \frac{{27}}{{169}} \cr & \Rightarrow x = 27 \cr} $$

$$\eqalign{ & = \sqrt {0.000441} \cr & = \sqrt {\frac{{441}}{{{{10}^6}}}} \cr & = \frac{{\sqrt {441} }}{{\sqrt {{{10}^6}} }} \cr & = \frac{{21}}{{{{10}^3}}} \cr & = \frac{{21}}{{1000}} \cr & = 0.021 \cr} $$

$$\eqalign{ & = {1.5^2} \times \sqrt {0.0225} \cr & = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr & = 2.25 \times \frac{{15}}{{100}} \cr & = 2.25 \times 0.15 \cr & = 0.3375 \cr} $$

Given expression,
$$ = \sqrt {\frac{1}{{100}}} + \sqrt {\frac{{81}}{{100}}} + \sqrt {\frac{{121}}{{100}}} + $$      $$\sqrt {\frac{9}{{10000}}} $$
$$\eqalign{ & = \frac{1}{{10}} + \frac{9}{{10}} + \frac{{11}}{{10}} + \frac{3}{{100}} \cr & = 0.1 + 0.9 + 1.1 + 0.03 \cr & = 2.13 \cr} $$

The first perfect square number after 50 is 64 $$\left( {64 = {8^2}} \right)$$   and the last perfect square number before 1000 is 961 $$\left[ {961 = {{\left( {31} \right)}^2}} \right]$$
So, the perfect squares between 50 and 1000 are the squares of numbers from 8 to 31.
(31 - 8) + 1 = 24
Clearly, these are 24 in number.

Clearly, the man was born between 1800 and 1850 is 1849.
$${\text{And, }}1849 = {\left( {43} \right)^2}$$
So, the man was 43 years old in 1849
$$\eqalign{ & {\text{Thus, he was born in }} \cr & = \left( {1849 - 43} \right) \cr & = 1806 \cr} $$

$$\eqalign{ & \,\,\,\,\,\,1|\overline 1 \,\,\overline {58} \,\,\overline {76} \,(126 \cr & \,\,\,\,\,\,\,\,\,|1 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,22|\,\,\,\,\,\,58 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,44 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & 246\,|\,\,\,\,\,\,\,14\,76 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,14\,76 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,x \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \therefore \sqrt {15876} = 126 \cr} $$

$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,1|\overline 3 \,.\,\,\overline {00} \,\,\overline {00} \,\,\overline {00} \,(1.732 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|1 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,27|\,\,2\,\,\,00 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,1\,\,\,89 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,343\,|\,\,\,\,\,\,\,\,\,11\,\,00 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,10\,\,29 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & 3492\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,71\,\,00 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,69\,\,84 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,16 \cr} $$
$$\eqalign{ & \therefore \sqrt 3 \cr & = 1.73 \cr & = \frac{{173}}{{100}} \cr} $$

$$\eqalign{ & {\text{Let}}, \cr & \sqrt {\frac{{16}}{{25}}} \times \sqrt {\frac{x}{{25}}} \times \frac{{16}}{{25}} = \frac{{256}}{{625}} \cr & {\text{Then}}, \cr & \frac{4}{5} \times \frac{{\sqrt x }}{5} \times \frac{{16}}{{25}} = \frac{{256}}{{625}} \cr & \Leftrightarrow \frac{{64\sqrt x }}{{625}} = \frac{{256}}{{625}} \cr & \Leftrightarrow \sqrt x = \frac{{256}}{{625}} \times \frac{{625}}{{64}} \cr & \Leftrightarrow x = {4^2} \cr & \Leftrightarrow x = 16 \cr} $$

$$\eqalign{ & {\text{Method 1:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow x = \sqrt {15 \times \left(15 \times 9 \right)} \cr & \Leftrightarrow x = \sqrt {{{15}^2} \times {3^2}} \cr & \Leftrightarrow x = 15 \times 3 \cr & \Leftrightarrow x = 45 \cr & \cr & {\text{Method 2:}} \cr & {\text{Let the missing number be }}x \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 15 \times 135 \cr & \Leftrightarrow {x^2} = 2025 \cr & \Leftrightarrow x = \sqrt {{2025} } \cr & \Leftrightarrow x = 45 \cr} $$

$$\eqalign{ & {\text{Let }}\frac{x}{{1776}} = \frac{{111}}{x} \cr & {\text{Then, }} \cr & \Leftrightarrow {x^2} = 111 \times 1776 \cr & \Leftrightarrow {x^2} = 111 \times 111 \times 16 \cr & \Leftrightarrow x = \sqrt {{{\left( {111} \right)}^2} \times {{\left( 4 \right)}^2}} \cr & \Leftrightarrow x = 111 \times 4 \cr & \Leftrightarrow x = 444 \cr} $$

$$\eqalign{ & {\text{Let,}} \cr & {\text{ }}\frac{{4\frac{1}{2}}}{x} = \frac{x}{{32}} \cr & {\text{Then,}} \cr & \Leftrightarrow {x^2} = 32 \times \frac{9}{2} \cr & \Leftrightarrow {x^2} = 144 \cr & \Leftrightarrow x = \sqrt {144} \cr & \Leftrightarrow x = 12 \cr} $$

$$\eqalign{ & \Leftrightarrow {{\text{n}}^2} = {\left( {{\text{25}}} \right)^{64}} \times {\left( {64} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {\left( {{{\text{5}}^2}} \right)^{64}} \times {\left( {{2^6}} \right)^{25}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{150}} \cr & \Leftrightarrow {{\text{n}}^2} = {5^{128}} \times {2^{128}} \times {2^{22}} \cr & \Leftrightarrow n = {5^{64}} \times {2^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {\left( {5 \times 2} \right)^{64}} \times {2^{11}} \cr & \Leftrightarrow n = {10^{64}} \times 2048 \cr} $$
∴ Sum of digits of n
= 2 + 0 + 4 + 8
= 14

$$\eqalign{ & \Leftrightarrow \frac{{0.13}}{{{p^2}}} = 13 \cr & \Leftrightarrow {p^2} = \frac{{0.13}}{{13}} \cr & \Leftrightarrow {p^2} = \frac{1}{{100}} \cr & \Leftrightarrow p = \sqrt {\frac{1}{{100}}} \cr & \Leftrightarrow p = \frac{1}{{10}} \cr & \Leftrightarrow p = 0.1 \cr} $$

$$\eqalign{ & \Leftrightarrow \sqrt {{3^n}} = 729 \cr & \Leftrightarrow \sqrt {{3^n}} = {3^6} \cr & \Leftrightarrow {\left( {\sqrt {{3^n}} } \right)^2} = {\left( {{3^6}} \right)^2} \cr & \Leftrightarrow {3^n} = {3^{12}} \cr & \Leftrightarrow n = 12 \cr} $$

$$\eqalign{ & = \sqrt {4{a^2} - 4a + 1} + 3a \cr & = \sqrt {{{\left( 1 \right)}^2} + {{\left( {2a} \right)}^2} - 2 \times 1 \times 2a} + 3a \cr & = \sqrt {{{\left( {1 - 2a} \right)}^2}} + 3a \cr & = \left( {1 - 2a} \right) + 3a \cr & = \left( {1 + a} \right) \cr & = \left( {1 + 0.1039} \right) \cr & = 1.1039 \cr} $$

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

$$\eqalign{ & {\text{Let, }} \cr & {\left( {15} \right)^2} + {\left( {18} \right)^2} - 20 = \sqrt x \cr & {\text{Then, }} \cr & \sqrt x = 225 + 324 - 20 \cr & \Leftrightarrow x = {\left( {529} \right)^2} \cr & \Leftrightarrow x = 279841 \cr} $$

$$\eqalign{ & \,\,\,\,\,2|4\,\,\overline {12} \,\,\overline {09} \,(203 \cr & \,\,\,\,\,\,\,\,\,|4 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,40|\,\,\,\,\,\,\,\,12 \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,0 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,12\,09 \cr & 403|\,\,\,\,\,\,\,\,\,12\,09 \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \cr & \,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,x \cr & \,\,\,\,\,\,\,\,\,| - - - - - - - - \, \cr & \therefore \sqrt {41209} = 203 \cr} $$

The number of digits of the square root of a perfect square number of n digits is
$$\eqalign{ & {\text{(i)}}\frac{n}{2}{\text{, if n is even}} \cr & {\text{(ii)}}\frac{{n + 1}}{2}{\text{, if n is odd}} \cr & {\text{Here, }}n = 12 \cr & {\text{So, required number of digits}} \cr & = \frac{n}{2} \cr & = \frac{{12}}{2} \cr & = 6{\text{ }} \cr} $$

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

$$\eqalign{ & \sqrt {4{a^2} - 4a + 1} + 3a \cr & = \sqrt {{{\left( 1 \right)}^2} + {{\left( {2a} \right)}^2} - 2 \times 1 \times 2a} + 3a \cr & = \sqrt {{{\left( {1 - 2a} \right)}^2}} + 3a \cr & = \left( {1 - 2a} \right) + 3a \cr & = \left( {1 + a} \right) \cr & = \left( {1 + 0.1039} \right) \cr & = 1.1039 \cr} $$

$$\eqalign{ & {\text{Given expression,}} \cr & \sqrt {1.30} + \sqrt {1300} + \sqrt {0.0130} \cr & = \sqrt {\frac{{130}}{{100}}} + \sqrt {13 \times 100} + \sqrt {\frac{{130}}{{10000}}} \cr & = \frac{{\sqrt {130} }}{{10}} + \sqrt {13} \times 10 + \frac{{\sqrt {130} }}{{100}} \cr & = \frac{{11.40}}{{10}} + 3.605 \times 10 + \frac{{11.40}}{{100}} \cr & = 1.14 + 36.05 + 0.114 \cr & = 37.304 \cr} $$

$$\eqalign{ & \Rightarrow {\text{Q}} = \sqrt {\frac{{{{\left( {8{\text{R}}} \right)}^2}}}{4}} \cr & \Rightarrow {\text{Q}} = \frac{{\sqrt {{{\left( {8{\text{R}}} \right)}^2}} }}{{\sqrt 4 }} \cr & \Rightarrow {\text{Q}} = \frac{{8{\text{R}}}}{2} \cr & \Rightarrow {\text{Q}} = 4{\text{R}} \cr} $$

The smallest such number is 1444 $$\left[ {1444 = {{\left( {38} \right)}^2}} \right]$$
It lies between 1000 and 2000.