Square And Qube

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

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

$${\text{Let}},$$
$$\sqrt x \times \sqrt {484} = 1034$$
$${\text{Then}},$$
$$\sqrt x \times 22 = 1034$$
$$ ⇒ \sqrt x = \frac{{1034}}{{22}} = 47$$
$$ ⇒ x = {\left( {47} \right)^2}$$
$$ ⇒ x = 2209$$

$${\text{Let}},$$
$$\sqrt {11881} \times \sqrt x = 10137$$
$${\text{Then}},$$
$$109 \times \sqrt x = 10137$$
$$ ⇒ \sqrt x = \frac{{10137}}{{109}} = 93$$
$$ ⇒ x = {\left( {93} \right)^2}$$
$$ ⇒ x = 8649$$

$$\frac{{4050}}{{\sqrt x }} = 450$$
$$ ⇒ \sqrt x = \frac{{4050}}{{450}}$$
$$ ⇒ \sqrt x = 9$$
$$ ⇒ x = {\left( 9 \right)^2}$$
$$ ⇒ x = 81$$

$$\,\,\,\,\,\,\,\,\,\,1|\overline 2 \,.\,\,\overline {00} \,\,\overline {00} \,\,\overline {00} \,(1.414$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,|1$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - $$
$$\,\,\,\,\,\,24|\,\,1\,00$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,96$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - $$
$$\,\,\,281\,|\,\,\,\,\,\,\,\,\,400$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,281$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - $$
$$2824\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11900$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11296$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - $$
$$\therefore \sqrt 2 = 1.414$$

$$ ⇒ 3\sqrt 5 + \sqrt {125} = 17.88$$
$$ ⇒ {\text{ }}3\sqrt 5 + \sqrt {25 \times 5} = 17.88$$
$$ ⇒ {\text{ }}3\sqrt 5 + 5\sqrt 5 = 17.88$$
$$ ⇒ {\text{ }}8\sqrt 5 = 17.88$$
$$ ⇒ \sqrt 5 = 2.235$$
$$\therefore \sqrt {80} + 6\sqrt 5 $$
$$ = \sqrt {16 \times 5} + 6\sqrt 5 $$
$$ = 4\sqrt 5 + 6\sqrt 5 $$
$$ = 10\sqrt 5 $$
$$ = \left( {10 \times 2.235} \right)$$
$$ = 22.35$$

Given expression,
$$\sqrt {4 \times 2} + 2\sqrt {16 \times 2} - 3\sqrt {64 \times 2} $$       $$ + $$ $$4\sqrt {25 \times 2} $$
$$ = 2\sqrt 2 + 8\sqrt 2 - 24\sqrt 2 + 20\sqrt 2 $$
$$ = 6\sqrt 2 $$
$$ = 6 \times 1.414$$
$$ = 8.484$$

$${\text{Given expression,}}$$
$$ = \frac{{3\sqrt {12} }}{{2\sqrt {28} }} \times \frac{{\sqrt {98} }}{{2\sqrt {21} }}$$
$$ = \frac{{3\sqrt {4 \times 3} }}{{2\sqrt {4 \times 7} }} \times \frac{{\sqrt {49 \times 2} }}{{2\sqrt {21} }}$$
$$ = \frac{{6\sqrt 3 }}{{4\sqrt 7 }} \times \frac{{7\sqrt 2 }}{{2\sqrt {21} }}$$
$$ = \frac{{21\sqrt 6 }}{{4\sqrt {7 \times 21} }}$$
$$ = \frac{{21\sqrt 6 }}{{28\sqrt 3 }}$$
$$ = \frac{3}{4}\sqrt 2 $$
$$ = \frac{3}{4} \times 1.414$$
$$ = 3 \times 0.3535$$
$$ = 1.0605$$

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