Class 8 Squares and Square Roots Exercise 3.8-1

\begin{array}{l} \text {Q1. Find the square root of each of the following correct to three places of decimal. } \\ (i) \quad 5 \quad (ii) \quad 7 \quad (iii) \quad 17 \\ \\ (iv) \quad 20 \quad (v) \quad 66 \quad (vi) \quad 427 \\ \\ (vii) \quad 1.7 \quad (viii) \quad 23.1 \quad (ix) \quad 2.5 \\ \\ (x) \quad 237.615 \quad (xi) \quad 15.3215 \quad (xii) \quad 0.9 \\ \\ (xiii) \quad 0.1 \quad (xiv) \quad 0.016 \quad (xv) \quad 0.00064 \\ \\ (xvi) \quad 0.019 \quad (xvii) \quad \frac{7}{8} \quad (xviii) \quad \frac{5}{12} \\ \\ (xix) \quad 2 \frac{1}{2} \quad (xx) \quad 287 \frac{5}{8} \\ \\\text {Sol. } (i) 5 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 2.236 \\ \hline 2 & \overline{5.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 4 \\ \hline 42 & ~100 \\ & ~84 \\ \hline 443 & ~~~1600 \\ & ~~~1329 \\ \hline 4466 & ~~~~~21700 \\ & ~~~~~26796 \\ \hline & ~~~~~~ 304 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{5} = 2.236 \\ \end{array}\begin{array}{l} \text {(ii) } 7 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 2.645 \\ \hline 2 & \overline{7.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 4 \\ \hline 46 & ~300 \\ & ~276 \\ \hline 524 & ~~~2400 \\ & ~~~2096 \\ \hline 5285 & ~~~~~30400 \\ & ~~~~~26425 \\ \hline & ~~~~~~ 3975 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{7} = 2.645 \\ \end{array}\begin{array}{l} (iii) 17 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 4.123 \\ \hline 4 & \overline{17.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 16 \\ \hline 81 & ~100 \\ & ~81 \\ \hline 822 & ~~~1900 \\ & ~~~1644 \\ \hline 8243 & ~~~~~25600 \\ & ~~~~~24729 \\ \hline & ~~~~~~ 871 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{17} = 4.123 \\ \end{array}\begin{array}{l} (iv) 20 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 4.472 \\ \hline 4 & \overline{20.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 16 \\ \hline 84 & ~400 \\ & ~336 \\ \hline 887 & ~~~6400 \\ & ~~~6209 \\ \hline 8942 & ~~~~~19100 \\ & ~~~~~17884 \\ \hline & ~~~~~~ 1216 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{20} = 4.472 \\ \end{array}\begin{array}{l} (v) 66 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 8.124 \\ \hline 8 & \overline{66.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 64 \\ \hline 161 & ~200 \\ & ~161 \\ \hline 1622 & ~~~3900 \\ & ~~~3244 \\ \hline 16244 & ~~~~~65600 \\ & ~~~~~54976 \\ \hline & ~~~~~~ 624 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{66} = 8.124 \\ \end{array}\begin{array}{l} \text {(vi) } 427 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 20.663 \\ \hline 2 & 4 ~ \overline{27.} ~ \overline{00} ~ \overline{00} ~ \overline{00} \\ & 4 \\ \hline 40 & ~027 \\ & ~0 \\ \hline 406 & ~~~2700 \\ & ~~~2436 \\ \hline 4126 & ~~~~~26400 \\ & ~~~~~24756 \\ \hline 41323 & ~~~~~164400 \\ & ~~~~~123969 \\ \hline & ~~~~~~ 40431 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{427} = 20.663 \\ \end{array}\begin{array}{l} \text {(vii) } 1.7 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 1.303 \\ \hline 1 & 1. ~ \overline{70} ~ \overline{00} ~ \overline{00} \\ & 1 \\ \hline 23 & ~70 \\ & ~69 \\ \hline 260 & ~~~100 \\ & ~~~0 \\ \hline 2603 & ~~~~~10000 \\ & ~~~~~7809 \\ \hline & ~~~~~~ 2191 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{1.7} = 1.303 \\ \end{array}\begin{array}{l} \text {(viii) } 23.1 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 4.806 \\ \hline 4 & \overline{23.} ~ \overline{10} ~ \overline{00} ~ \overline{00} \\ & 16 \\ \hline 88 & ~710 \\ & ~704 \\ \hline 960 & ~~~600 \\ & ~~~0 \\ \hline 96122 & ~~~~~236400 \\ & ~~~~~192244 \\ \hline & ~~~~~~ 44156 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{23.1} = 4.806 \\ \end{array}\begin{array}{l} \text {(ix) } 2.5 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 1.581 \\ \hline 1 & \overline{2.} ~ \overline{50} ~ \overline{00} ~ \overline{00} \\ & 1 \\ \hline 25 & ~150 \\ & ~125 \\ \hline 308 & ~~~2500 \\ & ~~~2464 \\ \hline 3161 & ~~~~~3600 \\ & ~~~~~3161 \\ \hline & ~~~~~~ 439 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{2.5} = 1.581 \\ \end{array}\begin{array}{l} \text {(x) } 237.615 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 15.414 \\ \hline 1 & 2 ~ \overline{37.} ~ \overline{61} ~ \overline{50} ~ \overline{00} \\ & 1 \\ \hline 25 & ~137 \\ & ~125 \\ \hline 304 & ~~~1261 \\ & ~~~1216 \\ \hline 3081 & ~~~~~4550 \\ & ~~~~~3081 \\ \hline 30824 & ~~~~~146900 \\ & ~~~~~123296 \\ \hline & ~~~~~~ 23604 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{237.615} = 15.414 \\ \end{array}\begin{array}{l} \text {(xi) } 15.3215 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 3.914 \\ \hline 3 & \overline{15.} ~ \overline{32} ~ \overline{15} ~ \overline{00} \\ & 9 \\ \hline 69 & ~632 \\ & ~621 \\ \hline 781 & ~~~1115 \\ & ~~~781 \\ \hline 7824 & ~~~~~33400 \\ & ~~~~~31295 \\ \hline & ~~~~~~ 2104 \\ \end{array}\begin{array}{l} \text {(xii) } 0.9 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.948 \\ \hline 0 & \overline{0.} ~ \overline{90} ~ \overline{00} ~ \overline{00} \\ & 0 \\ \hline 9 & ~90 \\ & ~81 \\ \hline 184 & ~~~900 \\ & ~~~736 \\ \hline 1888 & ~~~~~16400 \\ & ~~~~~15104 \\ \hline & ~~~~~~ 1296 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{0.9} = 0.948 \\ \end{array}\begin{array}{l} \text {(xiii) } 0.1 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.316 \\ \hline 0 & \overline{0.} ~ \overline{10} ~ \overline{00} ~ \overline{00} \\ & 0 \\ \hline 3 & ~10 \\ & ~9 \\ \hline 61 & ~~~100 \\ & ~~~61 \\ \hline 626 & ~~~~~3900 \\ & ~~~~~3756 \\ \hline & ~~~~~~ 144 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{0.1} = 0.316 \\ \end{array}\begin{array}{l} \text {(xiv) } 0.016 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.126 \\ \hline 0 & \overline{0.} ~ \overline{01} ~ \overline{60} ~ \overline{00} \\ & 0 \\ \hline 1 & ~01 \\ & ~1 \\ \hline 22 & ~~~60 \\ & ~~~44 \\ \hline 246 & ~~~~~1600 \\ & ~~~~~1476 \\ \hline & ~~~~~~ 124 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{0.016} = 0.126 \\ \end{array}\begin{array}{l} \text {(xv) } 0.00064 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.0252 \\ \hline 0 & \overline{0.} ~ \overline{00} ~ \overline{06} ~ \overline{40} \\ & 0 \\ \hline 2 & ~006 \\ & ~4 \\ \hline 45 & ~~~240 \\ & ~~~225 \\ \hline 502 & ~~~~~1500 \\ & ~~~~~1004 \\ \hline & ~~~~~~ 496 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{0.00064} = 0.0252 \\ \end{array}\begin{array}{l} \text {(xvi) } 0.019 \\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.137 \\ \hline 0 & \overline{0.} ~ \overline{01} ~ \overline{90} ~ \overline{00} \\ & 0 \\ \hline 1 & ~01 \\ & ~1 \\ \hline 23 & ~~~90 \\ & ~~~69 \\ \hline 267 & ~~~~~2100 \\ & ~~~~~1869 \\ \hline & ~~~~~~ 231 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{0.019} = 0.137 \\ \end{array}\begin{array}{l} \text {(xvii) } \frac {7}{8} = 0.875\\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.9354 \\ \hline 0 & \overline{0.} ~ \overline{87} ~ \overline{50} ~ \overline{00} \\ & 0 \\ \hline 9 & ~087 \\ & ~81 \\ \hline 183 & ~~~650 \\ & ~~~549 \\ \hline 1865 & ~~~~~10100 \\ & ~~~~~9325 \\ \hline & ~~~~~~ 775 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{\frac {7}{8}} = \sqrt{0.875} = 0.935 \\ \end{array}\begin{array}{l} \text {(xviii) } \frac {5}{12} = 0.416666\\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 0.645 \\ \hline 0 & \overline{0.} ~ \overline{41} ~ \overline{66} ~ \overline{66} \\ & 0 \\ \hline 6 & ~41 \\ & ~36 \\ \hline 124 & ~~~566 \\ & ~~~496 \\ \hline 1285 & ~~~~~7066 \\ & ~~~~~6245 \\ \hline & ~~~~~~ 641 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{\frac {5}{12}} = \sqrt{0.416666} = 0.645 \\ \end{array}\begin{array}{l} \text {(xix) } 2\frac {1}{2} = 2.5\\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 1.581\\ \hline 1 & \overline{2.} ~ \overline{50} ~ \overline{00} ~ \overline{00} \\ & 1 \\ \hline 25 & ~150 \\ & ~125 \\ \hline 308 & ~~~2500 \\ & ~~~2464 \\ \hline 3161 & ~~~~~3600 \\ & ~~~~~3161 \\ \hline & ~~~~~~ 439 \\ \end{array}\begin{array}{l} \therefore \quad \sqrt{2\frac {1}{2}} = \sqrt{2.5} = 1.581 \\ \end{array}\begin{array}{l} \text {(xx) } 287\frac {5}{8} = 287.625\\ \text {By using long division method, we have } \\ \end{array}\begin{array}{l|l} & 16.959\\ \hline 1 & \overline{2} ~ \overline{87.} ~ \overline{62} ~ \overline{50} \\ & 1 \\ \hline 26 & ~187 \\ & ~156 \\ \hline 329 & ~~~3162 \\ & ~~~2961 \\ \hline 3385 & ~~~~~20150 \\ & ~~~~~16925 \\ \hline 33909 & ~~~~~~~322500 \\ & ~~~~~~~305181 \\ \hline & ~~~~~~~17319\\ \end{array}\begin{array}{l} \therefore \quad \sqrt{287\frac {5}{8}} = \sqrt{287.625} = 16.959 \\ \end{array}