\( 3^ { \frac{1}{2}} \times \sqrt 3\)
\( = 3^ { \frac{1}{2}} \times 3^ { \frac{1}{2}}\)
\( = {3^{\frac{1}{2} + \frac{1}{2}}}\)
\(=3^{\frac{1+1}{2}}\)
\(=3^{\frac{2}{2}}\)
\( = 3\)
সুতরাং নির্ণেয় গুণফল = 3.

নির্ণেয় সংখ্যা \(4 \div 2\sqrt 2 = \frac{4}{{2\sqrt 2 }}\)
\( = \frac{{4 \times \sqrt 2 }}{{2\sqrt 2 \times \sqrt 2 }}\)
\( = \frac{{4\sqrt 2 }}{{2 \times 2}}\)
\( = \frac{{4\sqrt 2 }}{4}\)
\( = \sqrt 2 \)
\(\therefore\) নির্ণেয় গুণফল \( = \sqrt 2 \)

\(3\sqrt 5 \times 5\sqrt 3 \)
\( = (3 \times 5) \times \sqrt 5 \times \sqrt 3 \)
\( = 15 \times \sqrt {5 \times 3} \)
\( = 15\sqrt {15} \)
\(\therefore\) নির্ণেয় গুণফল \( = 15\sqrt {15} \)

\(\sqrt{6} \times \sqrt{15}=x \sqrt{10}\)
\(\therefore x=\frac{\sqrt{6} \times \sqrt{15}}{\sqrt{10}}\)
\(=\frac{\sqrt{6 \times 15}}{\sqrt{10}}\)
\(=\frac{\sqrt{2 \times 3 \times 3 \times 5}}{\sqrt{10}}\)
\(=\frac{3 \sqrt{2 \times 5}}{\sqrt{2 \times 5}}=3 \)
\(\therefore x = 3 \)
\(\therefore\) নির্ণেয় \(x\) – এর মান \(= 3\)

\((\sqrt 5 + \sqrt 3 )(\sqrt 5 - \sqrt 3 ) = 25 - {x^2}\)
বা, \({(\sqrt 5 )^2} - {(\sqrt 3 )^2} = 25 - {x^2}\)
বা, \(5 - 3 = 25 - {x^2}\)
বা, \(2 = 25 - {x^2}\)
বা, \({x^2} = 25 - 2 = 23\)
বা, \(x^{2}=23\)
\(\therefore x = \pm \sqrt {23} \)
\(\therefore\) নির্ণেয় \(x\) – এর মান \(=\pm \sqrt{23}\)

\( \sqrt 7 \times \sqrt {14} \)
\(\sqrt 7 \times \sqrt 7 \times \sqrt 2 \)
\( = 7\sqrt 2 \)
\(\therefore\) নির্ণেয় গুনফল \(=7 \sqrt{2}\)

\( \sqrt {12} \times 2\sqrt 3 \)
\(=2\sqrt 3 \times 2\sqrt 3 \)
\(=2 \sqrt{3} \times 2 \sqrt{3}\)
\( = 4 \times 3 \)
\( = 12\)
\(\therefore\) নির্ণেয় গুনফল = 12

\( \sqrt 5 \times \sqrt {15} \times \sqrt 3 \)
\(\sqrt {5 \times 3 \times 15}\)
\( = \sqrt {15 \times 15} \)
\( = 15\)
\(\therefore\) নির্ণেয় গুনফল = 15

\(\sqrt 2 (3 + \sqrt 5 ) \)
\(=3 \sqrt{2}+\sqrt{5} \times \sqrt{2}\)
\( = 3\sqrt 2 + \sqrt {10} \)
\(\therefore\) নির্ণেয় গুনফল \( = 3\sqrt 2 + \sqrt {10} \)

\((\sqrt 2 + \sqrt 3 )(\sqrt 2 - \sqrt 3 )\)
\( = {(\sqrt 2 )^2} - {(\sqrt 3 )^2}\)
\(= 2 - 3 = - 1\)
\(\therefore\) নির্ণেয় গুণফল \(= -1\)

\((2\sqrt 3 + 3\sqrt 2 )(4\sqrt 2 + \sqrt 5 )\)
\( = 8\sqrt 6 + 12\sqrt 4 + 2\sqrt {15} + 3\sqrt {10} \)
\( = 8\sqrt 6 + 12 \times 2 + 2\sqrt {15} + 3\sqrt {10} \)
\( = 8\sqrt 6 + 24 + 2\sqrt {15} + 3\sqrt {10} \)
\(\therefore\) নির্ণের গুণফল \( = 8\sqrt 6 + 24 + 2\sqrt {15} + 3\sqrt {10} \)

\((\sqrt 3 + 1)(\sqrt 3 - 1)(2 - \sqrt 3 )(4 + 2\sqrt 3 )\)
\( = \{ (\sqrt 3 + 1)(\sqrt 3 - 1)\{ (2 - \sqrt 3 )(4 + 2\sqrt 3 )\} \)
\( = \left\{ {{{(\sqrt 3 )}^2} - {{(1)}^2}} \right\}(2 - \sqrt 3 )(4 + 2\sqrt 3 )\)
\( = (3 - 1)(2 - \sqrt 3 )(4 + 2\sqrt 3 )\)
\( = 2(2 - \sqrt 3 )(4 + 2\sqrt 3 )\)
\( = (4 - 2\sqrt 3 )(4 + 2\sqrt 3 )\)
\( = {(4)^2} - {(2\sqrt 3 )^2}\)
\( = 16 - 4 \times 3\)
\( = 16 - 12 = 4\)
\(\therefore\) নির্ণেয় গুনফল \(= 4\)

\(\sqrt 5 \) -এর করণী নিরসক উৎপাদক \(k\sqrt 5 \) (k একটি অশূন্য মূলদ সংখ্যা)
\(\therefore \sqrt x \equiv k\sqrt 5 \)
বা, \({\rm{ }}\sqrt x = 1.\sqrt 5 \) (k-এর মান 1)
\(\therefore x = 5\)
\(\therefore\) \(x\)-এর মান \(x = 5\)

\(3\sqrt 2 \div 3 \)
\(=\frac{3 \sqrt{2}}{3}\)
\( = \sqrt 2 \)

\(7 \div \sqrt {48} \)
\( = \frac{7}{{\sqrt {48} }}\)
\( = \frac{7}{{\sqrt {4 \times 4 \times 3} }}\)
\( = \frac{7}{{4\sqrt 3 }}\)
এখানে লব বা হরকে ন্যূনতম \(\sqrt 3 \) দিয়ে গুণ করলে ভাগফলটি মূলদ হবে।
\(\therefore\) নির্ণেয় উত্তর \( = \sqrt 3 \)

\((\sqrt 5 + 2)\) )-এর অনুবন্ধী করণী \( = 2 - \sqrt 5 \) (উত্তর)।

\(\frac{{\sqrt 5 + \sqrt 2 }}{{\sqrt 7 }} = \frac{1}{7}(\sqrt {35} + a)\)
বা, \(\frac{{(\sqrt 5 + \sqrt 2 )\sqrt 7 }}{{\sqrt 7 \times \sqrt 7 }} = \frac{{\sqrt {35} + a}}{7}\)
বামদিকে লব ও হরকে \(\sqrt 7 \) দিয়ে গুণ করলে, \(\frac{{(\sqrt 5 + \sqrt 2 )\sqrt 7 }}{{\sqrt 7 \times \sqrt 7 }}\) \(= \frac{{\sqrt {35} + a}}{7}\)
বা, \(\frac{\sqrt{35}+\sqrt{14}}{7}=\frac{\sqrt{35}+a}{7}\)
বা, \(\sqrt {35} + \sqrt {14} = \sqrt {35} + a\)
বা, \(a = \sqrt {14} \)
\(\therefore a = \sqrt {14} \quad \)
\(\therefore\) \(a\)-এর মান \(=\sqrt{14}\)

\(\frac{5}{{\sqrt 3 - 2}}\) ভগ্নাংশের হরের একটি করণী নিরসক উৎপাদক \( = \sqrt 3 + 2\)

\(9 - 4\sqrt 5 \) এর অনুবন্ধী করণী \(9 + 4\sqrt 5 \)
\( - 2 - \sqrt 7 \) এর অনুবন্ধী করণী \( - 2 + \sqrt 7 \)

(i)\( \sqrt 5 + \sqrt 2 \) এর 2টি করণী নিরসক উৎপাদক হল \(\sqrt 5 - \sqrt 2 , - \sqrt 5 + \sqrt 2 \)
(ii)\(13 + \sqrt 6 \) এর 2টি করণী নিরসক উৎপাদক হল \(13 - \sqrt 6 , - 13 + \sqrt 6 \)
(iii)\(\sqrt 8 - 3\) এর 2টি করণী নিরসক উৎপাদক হল \( - \sqrt 8 - 3,\sqrt 8 + 3\)
(iv)\(\sqrt {17} - \sqrt {15}\) এর 2টি করণী নিরসক উৎপাদক হল \(\sqrt {17} + \sqrt {15}, - \sqrt {17} - \sqrt {15}\)

(i) \(\frac{{2\sqrt 3 + 3\sqrt 2 }}{{\sqrt 6 }}\)
\(= \frac{{(2\sqrt 3 + 3\sqrt 2 ) \times \sqrt 6 }}{{\sqrt 6 \times \sqrt 6 }}\)
\( = \frac{{2\sqrt {18} + 3\sqrt {12} }}{6}\)
\(= \frac{{6\sqrt 2 + 6\sqrt 3 }}{6}\)
\(=\frac{2 \sqrt{9 \times 2}+3 \sqrt{4 \times 3}}{6}\)
\(=\frac{2 \times 3 \sqrt{2}+3 \times 2 \sqrt{3}}{6}\)
\( = \frac{{6(\sqrt 2 + \sqrt 3 )}}{6}\) \(=\sqrt 2 + \sqrt 3 \)
(ii) \({\rm{ }}\frac{{\sqrt 2 - 1 + \sqrt 6 }}{{\sqrt 5 }}\)
\( = \frac{{(\sqrt 2 - 1 + \sqrt 6 ) \times \sqrt 5 }}{{\sqrt 5 \times \sqrt 5 }}\)
= \(\frac{{\sqrt {10} - \sqrt 5 + \sqrt {30} }}{5}\)
(iii) \(\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}\)
\(= \frac{{(\sqrt 3 + 1)(\sqrt 3 + 1)}}{{(\sqrt 3 - 1)(\sqrt 3 + 1)}}\)
\(=\frac{3+1+\sqrt{3}+\sqrt{3}}{(\sqrt{3})^{2}-(1)^{2}}\)
= \(\frac{{3 + 1 + 2\sqrt 3 }}{{{{(\sqrt 3 )}^2} - {{(1)}^2}}}\)
= \(\frac{{4 + 2\sqrt 3 }}{{3 - 1}}\)
= \(\frac{{4 + 2\sqrt 3 }}{2}\)
= \(\frac{{2(2 + \sqrt 3 )}}{2}\)
\( = 2 + \sqrt 3 \)
(iv) \(\frac{{3 + \sqrt 5 }}{{\sqrt 7 - \sqrt 3 }}\)
= \(\frac{{(3 + \sqrt 5 )(\sqrt 7 + \sqrt 3 )}}{{(\sqrt 7 - \sqrt 3 )(\sqrt 7 + \sqrt 3 )}}\)
= \(\frac{{3\sqrt 7 + \sqrt {35} + 3\sqrt 3 + \sqrt {15} }}{{{{(\sqrt 7 )}^2} - {{(\sqrt 3 )}^2}}}\)
\( = \frac{{3\sqrt 7 + \sqrt {35} + 3\sqrt 3 + \sqrt {15} }}{{7 - 3}}\)
= \(\frac{{3\sqrt 7 + \sqrt {35} + 3\sqrt 3 + \sqrt {15} }}{4}\)
(v) \(\frac{{3\sqrt 2 + 1}}{{2\sqrt 5 - 1}}\)
= \(\frac{{(3\sqrt 2 + 1)(2\sqrt 5 + 1)}}{{(2\sqrt 5 - 1)(2\sqrt 5 + 1)}}\)
= \(\frac{{6\sqrt {10} + 3\sqrt 2 + 2\sqrt 5 + 1}}{{{{(2\sqrt 5 )}^2} - {{(1)}^2}}}\)
\( = \frac{{6\sqrt {10} + 3\sqrt 2 + 2\sqrt 5 + 1}}{{20 - 1}}\)
= \(\frac{{6\sqrt {10} + 3\sqrt 2 + 2\sqrt 5 + 1}}{{19}}\)
(vi) \(\frac{{3\sqrt 2 + 2\sqrt 3 }}{{3\sqrt 2 - 2\sqrt 3 }}\)
= \(\frac{{(3\sqrt 2 + 2\sqrt 3 )(3\sqrt 2 + 2\sqrt 3 )}}{{(3\sqrt 2 - 2\sqrt 3 )(3\sqrt 2 + 2\sqrt 3 )}}\)
\(=\frac{18+6 \sqrt{6}+6 \sqrt{6}+12}{(3 \sqrt{2})^{2}-(2 \sqrt{3})^{2}}\)
= \(\frac{{18 + 12 + 12\sqrt 6 }}{{{{(3\sqrt 2 )}^2} - {{(2\sqrt 3 )}^2}}}\)
\( = \frac{{30 + 12\sqrt 6 }}{{18 - 12}} \)
\(= \frac{{30 + 12\sqrt 6 }}{6} \)
\(=\frac{6(5+2 \sqrt{6})}{6}\)
\(= 5 + 2\sqrt 6 \)

(i) \(\frac{{3\sqrt 2 + \sqrt 5 }}{{\sqrt 2 + 1}}\)
\(= \frac{{(3\sqrt 2 + \sqrt 5 )(\sqrt 2 - 1)}}{{(\sqrt 2 + 1)(\sqrt 2 - 1)}}\)
\(= \frac{{6 + \sqrt {10} - 3\sqrt 2 - \sqrt 5 }}{{{{(\sqrt 2 )}^2} - {{(1)}^2}}}\)
\(=\frac{6+\sqrt{10}-3 \sqrt{2}-\sqrt{5}}{2-1}\)
\(= \frac{{6 + \sqrt {10} - 3\sqrt 2 - \sqrt 5 }}{1}\)
\( = 6 + \sqrt {10} - 3\sqrt 2 - \sqrt 5\)
(ii) \(\frac{{2\sqrt 3 - \sqrt 2 }}{{\sqrt 2 - \sqrt 3 }}\)
\(= \frac{{(2\sqrt 3 - \sqrt 2 )(\sqrt 2 + \sqrt 3 )}}{{(\sqrt 2 + \sqrt 3 )(\sqrt 2 - \sqrt 3 )}} \)
\(= \frac{{2\sqrt 6 - 2 + 6 - \sqrt 6 }}{{{{(\sqrt 2 )}^2} - {{(\sqrt 3 )}^2}}}\)
\(=\frac{\sqrt{6}+4}{2-3}\)
\(= \frac{{\sqrt 6 + 4}}{{ - 1}} = - (4 + \sqrt 6 )\)
(iii) \(\frac{{3 + \sqrt 6 }}{{\sqrt 3 + \sqrt 2 }}\)
\(= \frac{{(3 + \sqrt 6 )(\sqrt 3 - \sqrt 2 )}}{{(\sqrt 3 + \sqrt 2 )(\sqrt 3 - \sqrt 2 )}}\)
= \(\frac{{3\sqrt 3 + 3\sqrt 2 - 3\sqrt 2 - 2\sqrt 3 }}{{{{(\sqrt 3 )}^2} - {{(\sqrt 2 )}^2}}}\)
\(=\frac{\sqrt{3}}{3-2}\)
\(= \frac{{\sqrt 3 }}{1}\)
\( = \sqrt 3\)

(i) \(\frac{{2\sqrt 5 + 1}}{{\sqrt 5 + 1}} - \frac{{4\sqrt 5 - 1}}{{\sqrt 5 - 1}}\)
\(= \frac{{(2\sqrt 5 + 1)(\sqrt 5 - 1) - (\sqrt 5 + 1)(4\sqrt 5 - 1)}}{{(\sqrt 5 + 1)(\sqrt 5 - 1)}}\)
\(=\frac{10+\sqrt{5}-2 \sqrt{5}-1-20-4 \sqrt{5}+\sqrt{5}+1}{(\sqrt{5})^{2}-(1)^{2}}\)
\( = \frac{{10 + \sqrt 5 - 2\sqrt 5 - 1 - 20 - 4\sqrt 5 + \sqrt 5 + 1}}{{5 - 1}}\)
\(= \frac{{ - 10 - 4\sqrt 5 }}{4}\)
\(=\frac{2(-5-2 \sqrt{5})}{4}\)
\(= \frac{{ - 5 - 2\sqrt 5 }}{2}\)
\(= - \frac{{(5 + 2\sqrt 5 )}}{2}\)
\(\therefore\) নির্ণেয় মান \(=-\frac{(5+2 \sqrt{5})}{2}\)
(ii) \(\frac{{8 + 3\sqrt 2 }}{{3 + \sqrt 5 }} - \frac{{8 - 3\sqrt 2 }}{{3 - \sqrt 5 }}\)
\(= \frac{{(8 + 3\sqrt 2 )(3 - \sqrt 5 ) - (8 - 3\sqrt 2 )(3 + \sqrt 5 )}}{{(3 + \sqrt 5 )(3 - \sqrt 5 )}}\)
\(=\frac{(8+3 \sqrt{2})(3-\sqrt{5})-(8-3 \sqrt{2})(3+\sqrt{5})}{(3)^{2}-(\sqrt{5})^{2}}\)
\( = \frac{{24 + 9\sqrt 2 - 8\sqrt 5 - 3\sqrt {10} - 24 - 8\sqrt 5 + 9\sqrt {2}-3\sqrt 10 }}{{9 - 5}}\)
\(= \frac{{18\sqrt 2 - 16\sqrt 5 }}{4}\)
\(=\frac{2(9 \sqrt{2}-8 \sqrt{5})}{4}\)
\( = \frac{{9\sqrt 2 - 8\sqrt 5 }}{2}\)
\(\therefore\) নির্ণেয় মান \(=\frac{9 \sqrt{2}-8 \sqrt{5}}{2}\)