\[? = 5 - [4 - {3 - (3- 3 - 6)}]\]

\[= 5 - [4 - { 3 - (- 6)}]\]

\[= 5 - [4 - {3 + 6}]\]

\[= 5 - [4 - 9]\]

\[= 5 + 5\]

\[= 10\]

\[\Bigg(\sqrt{2}+\frac{1}{\sqrt{2}}\Bigg)^2\]

\[=(\sqrt{2})^2+\Bigg(\frac{1}{\sqrt{2}}\Bigg)^2+2(\sqrt{2})\Bigg(\frac{1}{\sqrt{2}}\Bigg)\]

\[=2+\frac{1}{2}+2\]

\[=4+\frac{1}{2}\]

\[=4\frac{1}{2}\]

\[100 × 10 – 100 +2000 ÷100\]

\[= 100 × 10 – 100 + 20\]

\[= 100 (10 – 1) + 20\]

\[= 100 × 9 + 20\]

\[= 900 + 20\]

\[= 920\]

\[\frac{1120}{\sqrt{p}}=80\]

\[\Rightarrow 80\times\sqrt{p}=1120́\]

\[\Rightarrow \sqrt{p}=\frac{1120}{80}=14\]

\[\Rightarrow p=(14)^2=194\]

\[\frac{9}{20}-\left[\frac{1}{5}+\left\{\frac{1}{4}+\left(\frac{5}{6}-\overline{\frac{1}{3}+\frac{1}{2}}\right)\right\}\right]\]

\[=\frac{9}{20}-\left[\frac{1}{5}+\left\{\frac{1}{4}+\left(\frac{5}{6}-\frac{5}{6}\right)\right\}\right]\]

\[=\frac{9}{20}-\left[\frac{1}{5}+\frac{1}{4}\right]\]

\[=\frac{9}{20}-\frac{9}{20}\]

\[=0\]

Let \[\ast = x\], then

\[\left[\frac{\left(x\right)}{21}\times\frac{\left(x\right)}{189}\right]=1\]

\[(x)^2 = 21 × 189\]

\[\Rightarrow x=\sqrt{21\times189}=63\]

\[\left(\frac{5}{2}+\frac{3}{2}\right)\left(\frac{25}{4}-\frac{15}{4}+\frac{9}{4}\right)\]

\[=4\times\frac{19}{4}=19\]

\[=\frac{20\div5}{9+1}=\frac{4}{10}\]

\[=\frac{2}{5}\]

Let the value of \[(\ast) = x\].

\[\therefore \frac{50}{x}=\frac{x}{12\frac{1}{12}}\]

\[\Rightarrow 2x^2=50\times25\]

\[\Rightarrow x^2=25\times25\]

\[\therefore x=25\]

Expression \[=1+\frac{1}{1+\frac{1}{2}}\]

\[1+\frac{1}{\frac{2+1}{2}}\]

\[=1+\frac{2}{3}\]

\[=\frac{3+2}{3}\]

\[=\frac{5}{3}\]

\[\sqrt{0.000441}\]

\[= \sqrt{0.021\times0.021}\]

\[= 0.021\]

Expression \[=\frac{\sqrt{0.441}}{\sqrt{0.625}}\]

\[=\sqrt{\frac{441}{625}}\]

\[=\frac{21}{25}\]

\[=0.84\]

\[0.\overline{4}=\frac{4}{9}\]

\[\therefore\sqrt{\frac{4}{9}}\]

\[=\frac{2}{3}\]

\[=\frac{2\ \times3}{3\times3}\]

\[=\frac{6}{9}\]

\[=0.\overline{6}\]

\[\therefore\sqrt{64009}=253\]

\[a = 64\] and \[b = 289\]

\[\therefore a = \sqrt{64} = 8\]

and \[b = \sqrt{289} = 17\]

\[\therefore \left(\sqrt{\sqrt{a}+\sqrt{b}}-\sqrt{\sqrt{b}-\sqrt{a}}\right)^{\frac{1}{2}}\]

\[=\left(\sqrt{8+17}-\sqrt{17-8}\right)^{\frac{1}{2}}\]

\[=\left(\sqrt{25}-\sqrt{9}\right)^{\frac{1}{2}}\]

\[=\left(5-3\right)^{\frac{1}{2}}\]

\[=\left(2\right)^{\frac{1}{2}}\]

Look at the pattern :

\[1001 × 1001 = 1002001\]

\[1001×1001 × 1001 = 1003003001\]

\[\sqrt{x} \div \sqrt{441}=0.02\]

\[\Rightarrow \sqrt{x}=0.02\times21\]

\[\Rightarrow x=0.1764\]

\[\frac{\left(20^2-15^2\right)}{10^2}\]

\[=\frac{\left(20+15\right)\left(20-15\right)}{100}\]

\[=\frac{\left(35\times5\right)}{100}\]

\[=\frac{35}{20}\]

\[=\frac{7}{4}\]

\[=\frac{2}{3}\times\frac{3}{\frac{5}{6}\div\frac{10}{12}}\]

\[=\frac{2}{3}\times\frac{3}{\frac{5}{6}\times\frac{12}{10}}\]

\[=\frac{2}{3}\times\frac{3}{\frac{5}{6}\times\frac{6}{5}}\]

\[=\frac{2}{3}\times3\]

\[=2\]