বৰ্গ আৰু বৰ্গমূল।Class 8 Maths Exercise 6 (C)।Sankardev Sishu Niketan ।শংকৰদেৱ শিশু নিকেটন।Chapter 8। গণিত

 
Exercise – 6(C)
 
1. তলৰ সমীকৰণ বৰ্গমূলত কেইটাকৈ অংক থাকিব 
(i) 81                           (ii) 169                        (iii) 4489       
Ans- এটা                    Ans- দুটা                      Ans- দুটা    
(iv) 53361                 (v) 4553956
Ans- তিনিটা               Ans- চাৰিটা   
2. হৰণ পদ্ধথিৰে বৰ্গমূল নিৰ্ণয় কৰা —–
(i)  529
Solution:-
\[ \begin{array}{r|l}\ & 21\quad \\\hline 2 & 441 \quad \\& 4 \quad \\\hline41 & \underline{041} \quad \\& 041\\\hline\\ & 0\\\end{array} \]
  

(ii) 1296

Solution:- 

\[ \begin{array}{r|l}\ & 36\quad \\\hline 3 & 1296 \quad \\& 09\quad \\\hline66 & \underline{396} \quad \\& 396\\\hline\\ & 0\\\end{array} \]

(iii) 4096

Solution:- 

\[ \begin{array}{r|l}\ & 64\quad \\\hline 6 & 4096 \quad \\& 36\quad \\\hline124 & \underline{496} \quad \\& 496\\\hline\\ & 0\\\end{array} \]

(iv) 15129

Solution:- 

\[ \begin{array}{r|l}\ & 123\quad \\\hline 1 & 4096 \quad \\& 36\quad \\\hline22 & \underline{051} \quad \\& 044\\\hline243 & \underline{729} \quad \\& 729\\\hline\\ & 0\\\end{array} \]

(v) 11449

Solution:- 

\[ \begin{array}{r|l}\ & 107\quad \\\hline 1 & 11449 \quad \\& 1\quad \\\hline207 & \underline{11449} \quad \\& 11449\\\hline\\ & 0\\\end{array} \]

(vi) 361201

Solution:-

\[ \begin{array}{r|l}\ & 601\quad \\\hline 6 & 361201 \quad \\& 36\quad \\\hline1201 & \underline{1201} \quad \\& 1201\\\hline\\ & 0\\\end{array} \]

(vii) 54756

Solution:-

\[ \begin{array}{r|l}\ & 234\quad \\\hline 2 & 54756 \quad \\& 4\quad \\\hline43 & \underline{147} \quad \\& 129\\\hline464 & \underline{1856} \quad \\& 1856\\\hline\\ & 0\\\end{array} \]

(viii) 2271049

Solution:-

\[ \begin{array}{r|l}\ & 1507\quad \\\hline 1 & 2271049 \quad \\& 1\quad \\\hline25 & \underline{127} \quad \\& 125\\\hline3007 & \underline{21049} \quad \\& 21049\\\hline\\ & 0\\\end{array} \]

3. বৰ্গমূল উলিওৱা—–

(i) 7.29

Solution:- 

\[ \begin{array}{r|l}\ & 2.7\quad \\\hline 2 & 7.29 \quad \\& 4 \quad \\\hline47 & \underline{329} \quad \\& 329\\\hline\\ & 0\\\end{array} \]
 
(ii) 0.000009
Solution:- 
=\(\frac{9}{1000000}\)
 
=\(\sqrt{\frac{9}{1000000}}\)
 
= \(\frac{3}{1000}\)
 
(ii) 31.36
Solution:- 
=\(\frac{3136}{100}\)
 
=\(\sqrt{\frac{3136}{100}}\)
 
= \(\frac{56}{10}\)
 
(iv) 22.1841
Solution:- 
=\(\frac{221841}{10000}\)
 
=\(\sqrt{\frac{221841}{10000}}\)
 
= \(\frac{471}{100}\)
 
(v) 1.012036
Solution:- 
=\(\frac{1012036}{1000000}\)
 
=\(\sqrt{\frac{1012036}{1000000}}\)
 
= \(\frac{1006}{1000}\)
 
(f)  412.658596
Solution:- 
=\(\frac{412658596}{1000000}\)
 
=\(\sqrt{\frac{412658596}{1000000}}\)
 
= \(\frac{20314}{1000}\)
 
(g) \(\frac{1.69}{17.64}\)
 
Solution:- 
=\(\frac{1.69}{17.64}\)
 
=\(\frac{169}{1764}\)
 
=\(\sqrt{\frac{13^2}{42^2}}\)
 
= \(\frac{13}{42}\)
 
(h) \(\frac{0.36}{0.000004}\)
 
Solution:- 
=\(\frac{0.36}{0.000004}\)
 
=\(\frac{\frac{36}{100}}{\frac{4}{1000000}}\)
 
= \(\frac{36}{100}\)×\(\frac{1000000}{4}\)
 
= 90000
=\(\sqrt{300^2}\)
= 300
4. বন্ধনীত উল্লেখ কৰা সংখ্যাক দশমিক স্থানলৈ বৰ্গমূল নিৰ্ণয় কৰা ——
 
(i) 5 [4]
 

Solution:-

\[ \begin{array}{r|l}\ & 2.2360\quad \\\hline 2 & 5.00000000 \quad \\& 4\quad \\\hline42 & \underline{100} \quad \\& 084\\\hline443 & \underline{1600} \quad \\& 1329\\\hline4466 & \underline{27100} \quad \\& 26796\\\hline44720 & \underline{304000} \quad \\& \\\hline\\ & 0\\\end{array} \]

(ii) 1.7 [3]

Solution:- 

\[ \begin{array}{r|l}\ & 1.304\quad \\\hline 1 & 1.70000000 \quad \\& 1\quad \\\hline239 & \underline{070} \quad \\& 069\\\hline2604& \underline{10000} \quad \\& 09416\\\hline\\ & \\\end{array} \]

(iii) 5.652 [3]

Solution:- 

\[ \begin{array}{r|l}\ & 2.378\quad \\\hline 2 & 5.65200000 \quad \\& 4\quad \\\hline43 & \underline{165} \quad \\& 129\\\hline467& \underline{3620} \quad \\& 3269\\\hline4747& \underline{36100} \quad \\& 33229\\\hline\\ & \\\end{array} \]

(iv)  1.092 [5]

Solution:-

\[ \begin{array}{r|l}\ & 3.901\quad \\\hline 3 & 13.0000 \quad \\& 9\quad \\\hline39 & \underline{400} \quad \\& 351\\\hline4801& \underline{4900} \quad \\& 4801\\\hline& \underline{0099} \quad \\&\\&\ {}\\\end{array} \]

(vi) 22

Solution:- 

(i) 5 [4]
 

Solution:-

\[ \begin{array}{r|l}\ & 4.6904\quad \\\hline 4 & 22.000000 \quad \\& 16\quad \\\hline86 & \underline{600} \quad \\& 51`6\\\hline929 & \underline{8400} \quad \\& 8361\\\hline93804 & \underline{390000} \quad \\& 375216\\\hline & \underline{014784} \quad \\& \\\ & \\\end{array} \]

(viii) 300 [7]

Solution:- 

\[ \begin{array}{r|l}\ & 17.32050807\quad \\\hline 1 & 300.0000000000000000 \quad \\& 1\quad \\\hline27 & \underline{200} \quad \\& 189\\\hline343 & \underline{1100} \quad \\& 1029\\\hline3462 & \underline{7100} \quad \\& 6924\\\hline 346405& \underline{1760000} \quad \\& 1732025\\\hline 34641008& \underline{279750000} \quad \\& 263048064\\\hline 3464101607& \underline{167019360000} \quad \\&024248711249 \\\ & \\\end{array} \]

5. কি সৰু সংখ্যা বিয়োগ কৰিলে লতৰ সংখ্যাটোৰ ওচৰৰ বৰ্গ সংখ্যা পোৱা যাব ।  বৰ্গমূল উলিওৱা ——

(i) 825

Solution:- 

\[ \begin{array}{r|l}\ & 28\quad \\\hline 2 & 825 \quad \\& 4\quad \\\hline48 & \underline{425} \quad \\& 384\\\hline\\ & 041\\\end{array} \]

এতেকে বিয়োগকৰিবলগীয়া সংখ্যাটো হৈছে  41

∴ \(\sqrt{784}\) = 28

(ii) 1980

Solution:- 

\[ \begin{array}{r|l}\ & 44\quad \\\hline 4 & 1989 \quad \\& 4\quad \\\hline84 & \underline{389} \quad \\& 336\\\hline\\ & 053\\\end{array} \]

এতেকে বিয়োগকৰিবলগীয়া সংখ্যাটো হৈছে  53

∴ \(\sqrt{1936}\) = 44

(iii) 5607 

Solution:- 

\[ \begin{array}{r|l}\ & 75\quad \\\hline 7 & 5607\quad \\& 49\quad \\\hline144 & \underline{707} \quad \\& 576\\\hline\\ & 131`\\\end{array} \]

এতেকে বিয়োগকৰিবলগীয়া সংখ্যাটো হৈছে  131

∴ \(\sqrt{5476}\) = 74

(iv) 46666
Solution:- 
 

Solution:- 

\[ \begin{array}{r|l}\ & 216\quad \\\hline 2 & 46666\quad \\& 4\quad \\\hline41 & \underline{66} \quad \\& 41\\\hline426 & \underline{2566} \quad \\& 2556\\\hline\\ & 10`\\\end{array} \]

এতেকে বিয়োগকৰিবলগীয়া সংখ্যাটো হৈছে  10

∴ \(\sqrt{4656}\) = 216

6. আটাইতকৈ কি সৰু সংখ্যা যোগ কৰিলে তলৰ সংখ্যা ওচৰত একোটা বৰ্গ সংখ্যা পোৱা যাব ? বৰ্গমূল উলিওৱা। 

(i) 440 

Solution:-  440 ৰ লগত 1 যোগ কৰিলে বৰ্গ সংখ্যা পোৱা যায় ।

∴ \(\sqrt{441}\) = 21

(ii) 1220 

Solution:-  1220 ৰ লগত 5 যোগ কৰিলে বৰ্গ সংখ্যা পোৱা যায় ।

∴ \(\sqrt{1225}\) = 35

(iii) 8270

Solution:-  8270 ৰ লগত 11 যোগ কৰিলে বৰ্গ সংখ্যা পোৱা যায় ।

∴ \(\sqrt{8281}\) = 91

(iv) 40400

Solution:-  40400 ৰ লগত 1 যোগ কৰিলে বৰ্গ সংখ্যা পোৱা যায় ।

∴ \(\sqrt{40401}\) = 201

7. এখন বৰ্গাকাৰ ডবাবোৰ্ডৰ কালি 1024 বৰ্গ চে.মি । একোটা বাহুৰ দীঘ কিমান ?
 
Solution:-
\[ \begin{array}{r|l}\ & 32\quad \\\hline 3& 1024 \quad \\& 9 \quad \\\hline62 & \underline{124} \quad \\& 124\\\hline\\ & 0\\\end{array} \]
 
∴  বাহুৰ দীঘ = 32 cm
8. তালিকাৰ পৰা \(\sqrt{2}\) , \(\sqrt{3}\) , \(\sqrt{5}\),  আৰু \(\sqrt{7}\)ৰ মান ব্যৱহাৰ কৰি তলৰ সংখ্যাবোৰৰ বৰ্গমূল উলিোৱা ———-
(i) 10
 
Solution:-
= \(\sqrt{10}\)
= \(\sqrt{2×5}\) 
= \(\sqrt{2}×\sqrt{5}\)  
= 1.414 × 2.236
= 3.161
 
(ii) 63
 
Solution:-
= \(\sqrt{63}\)
= \(\sqrt{9×7}\) 
= \(\sqrt{3^2}×\sqrt{7}\)  
= 3 × 2.646
=  7,938
 
(iii)  \(\frac{3}{\sqrt{5}}\)
Solution:- 
= \(\frac{3}{2.236}\)
= 1.341
 
(iv) 2100
Solution:-
= \(\sqrt{2100}\) 
= \(\sqrt{3×7×10^2}\) 
= \(\sqrt{3}×\sqrt{7}\) × 10
= 1.732 × 2.646 × 10
= 45.828
9. তলৰ সংখ্যাকেইটাৰ নিকটম বৰ্গ সংখ্যাকেইটা কি ?
(i)  \(\sqrt{15}\) 
Ans:-  নিকটম বৰ্গ সংখ্যা 16
 
(ii)  \(\sqrt{46}\) 
Ans:-  নিকটম বৰ্গ সংখ্যা 49
 
(iii)  \(\sqrt{250}\) 
Ans:-  নিকটম বৰ্গ সংখ্যা 16
 
 
 
 
 
 
 
 
 
 
 

 

 

 

 

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