DAV Class 8 Maths Chapter 1 Worksheet 4
Squares and Square Roots Worksheet 4
1. Find the square root of the following numbers by long division method:
(i) 9801
\[ \begin{array}{r|l} & \phantom{00}99 \\ \hline 9 & \phantom{-}\overline{98} \,\, \overline{01} \\ & -81 \,\, \downarrow\\ \hline 189 & \phantom{-}17\,01 \\ & -17\,01 \\ \hline & \phantom{0000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{9801} = 99 \)
(ii) 6561
\[ \begin{array}{r|l} & \phantom{0}81 \\ \hline 8 & \phantom{-}\overline{65} \,\, \overline{61} \\ & -64 \,\, \downarrow\\ \hline 161 & \phantom{-0}161 \\ & -\phantom{0}161 \\ \hline & \phantom{0000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{6561} = 81 \)
(iii) 390625
\[ \begin{array}{r|l} & \phantom{0}625 \\ \hline 6 & \phantom{-}\overline{39} \,\, \overline{06} \,\, \overline{25} \\ & -36 \,\, \\ \hline 122 & \phantom{-0}306 \\ & -\phantom{0}244 \\ \hline 1245 & \phantom{-00}6225 \\ & -\phantom{00}6225 \\ \hline & \phantom{00000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{390625} = 625 \)
(iv) 108241
\[ \begin{array}{r|l} & \phantom{0}329 \\ \hline 3 & \phantom{-}\overline{10} \,\, \overline{82} \,\, \overline{41} \\ & \phantom{}-9 \\ \hline 62 & \phantom{0-}182 \\ & -\phantom{0}124 \\ \hline 649 & \phantom{00-}5841 \\ & -\phantom{00}5841 \\ \hline & \phantom{0000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{108241} = 329 \)
(v) 363609
\[ \begin{array}{r|l} & \phantom{00}603 \\ \hline 6 & \phantom{-}\overline{36} \,\, \overline{36} \,\, \overline{09} \\ & -36 \\ \hline 120 & \phantom{-0}036 \\ & -\phantom{00}00 \\ \hline 1203 & \phantom{-00}3609 \\ & -\phantom{00}3609 \\ \hline & \phantom{00000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{363609} = 603 \)
(vi) 120409
\[ \begin{array}{r|l} & \phantom{00}347 \\ \hline 3 & \phantom{-}\overline{12} \,\, \overline{04} \,\, \overline{09} \\ & -\phantom{0}9 \\ \hline 64 & \phantom{-0}304 \\ & -\phantom{0}256 \\ \hline 687 & \phantom{-00}4809 \\ & -\phantom{00}4809 \\ \hline & \phantom{00000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{120409} = 347 \)
(vii) 1471369
\[ \begin{array}{r|l} & \phantom{0}1213 \\ \hline 1 & \phantom{-}\overline{1} \,\, \overline{47} \,\, \overline{13} \,\, \overline{69} \\ & -1 \\ \hline 22 & \phantom{-}047 \\ & -\phantom{0}44 \\ \hline 241 & \phantom{-00}313 \\ & -\phantom{00}241 \\ \hline 2423 & \phantom{-000}7269 \\ & -\phantom{000}7269 \\ \hline & \phantom{000000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{1471369} = 1213 \)
(viii) 57121
\[ \begin{array}{r|l} & \phantom{0}239 \\ \hline 2 & \phantom{-}\overline{5} \,\, \overline{71} \,\, \overline{21} \\ & -4 \\ \hline 43 & \phantom{-}171 \\ & -129 \\ \hline 469 & \phantom{-0}4221 \\ & -\phantom{0}4221 \\ \hline & \phantom{00000}0 \\ \end{array} \]
Answer \( \color{red}\sqrt{57121} = 239 \)
2. Find the least number which must be subtracted from 6203 to obtain a perfect square. Also, find square root of the number so obtained.
\[ \begin{array}{r|l} & \phantom{0}78 \\ \hline 7 & \phantom{-}\overline{62} \,\, \overline{03} \\ & -49 \\ \hline 148 & \phantom{-}1303 \\ & -1184 \\ \hline & \phantom{0-}119\\ \end{array} \] \[ \begin{align*} \text{Required number} &= 119 \\ \text{Perfect square number} &= 6203 - 119 \\ &= 6084 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}78 \\ \hline 7 & \phantom{-}\overline{60} \,\, \overline{84} \\ & -49 \\ \hline 148 & \phantom{-}1184 \\ & -1184 \\ \hline & \phantom{00-}0\\ \end{array} \\ \] \[ \sqrt{6084} = 78 \]
Answer \( \color{red} 119, 78 \)
3. Find the greatest number of six digits which is a perfect square. Find the square root of this number.
Solution
\[ \begin{align*} \text{Greatest number of six digits} &= 999999 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}999 \\ \hline 9 & \phantom{-}\overline{99} \,\, \overline{99} \,\, \overline{99} \\ & -81 \\ \hline 189 & \phantom{-}1899 \\ & -1701 \\ \hline 1989 & \phantom{-0}19899 \\ & \phantom{}-17901 \\ \hline & \phantom{00-}1998 \\ \end{array} \] \[ \begin{align*} \text{Perfect square number} &= 999999 - 1998 \\ &= 998001 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}999 \\ \hline 9 & \phantom{-}\overline{99} \,\, \overline{80} \,\, \overline{01} \\ & -81 \\ \hline 189 & \phantom{-}1880 \\ & -1701 \\ \hline 1989 & \phantom{-0}17901 \\ & \phantom{}-17901 \\ \hline & \phantom{00000}0 \\ \end{array} \] \[ \sqrt{998001} = 999 \]
Answer \( \color{red} 998001, 999 \)
4. Find the least number which must be added to 6203 to obtain a perfect square. Also, find the square root of the number so obtained.
Solution
\[ \begin{array}{r|l} & \phantom{0}79 \\ \hline 7 & \phantom{-}\overline{62} \,\, \overline{03} \\ & -49 \\ \hline 149 & \phantom{-}1303 \\ & -1341 \\ \hline & \phantom{0}-38\\ \end{array} \] \[ \begin{align*} \text{Required number} &= 38 \\ \text{Perfect square number} &= 6203 + 38 \\ &= 6241 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}79 \\ \hline 7 & \phantom{-}\overline{62} \,\, \overline{41} \\ & -49 \\ \hline 149 & \phantom{-}1341 \\ & -1341 \\ \hline & \phantom{00-}0\\ \end{array} \] \[ \sqrt{6241} = 79 \]
Answer Least number \( = \color{red} 39 \) , \(\color{red} \sqrt{6241} = 79 \)
5. Find the least number of six digits which is a perfect square. Find the square root of this number.
Solution
\[ \begin{align*} \text{Least number of 6 digit} &= 100000 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}317 \\ \hline 3 & \phantom{-}\overline{10} \,\, \overline{00} \,\, \overline{00} \\ & -\phantom{1}9 \\ \hline 61 & \phantom{0-}100 \\ & - \phantom{00}61 \\ \hline 627 & \phantom{-00}3900 \\ & \phantom{0}-4389 \\ \hline & \phantom{00}-489 \\ \end{array} \] \[ \begin{align*} \text{Required number} &= 489 \\ \text{Perfect square number} &= 100000 + 489 \\ &= 100489 \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}317 \\ \hline 3 & \phantom{-}\overline{10} \,\, \overline{04} \,\, \overline{89} \\ & -\phantom{1}9 \\ \hline 61 & \phantom{0-}104 \\ & - \phantom{00}61 \\ \hline 627 & \phantom{-00}4389 \\ & \phantom{0}-4389 \\ \hline & \phantom{0--}0 \\ \end{array} \] \[ \sqrt{100489} = 317 \\ \]
Answer Perfect square number \( = \color{red} 100489 \) , \( \color{red} \sqrt{100489} = 317 \)
6. \(\sqrt{64432729} - \sqrt{9653449}\)
Solution
\[ \begin{array}{r|l} & \phantom{0}8027 \\ \hline 8 & \phantom{-}\overline{64} \,\, \overline{43} \,\, \overline{27} \,\, \overline{29} \\ & -64 \\ \hline 160 & \phantom{-0}043 \\ & -\phantom{-0}0 \\ \hline 1602& \phantom{-00}4327 \\ & -\phantom{00}3204 \\ \hline 16047 & \phantom{-00}112329 \\ & - \phantom{00}112729 \\ \hline & \phantom{000000}0 \\ \end{array} \] \[ \color{green}\sqrt{64432729} = 8027 \] \[ \begin{array}{r|l} & \phantom{0}3107 \\ \hline 3 & \phantom{-}\overline{9} \,\, \overline{65} \,\, \overline{34} \,\, \overline{49} \\ & -9 \\ \hline 61 & \phantom{-}065 \\ & -\phantom{0}61 \\ \hline 620 & \phantom{-00}434 \\ & -\phantom{00}000 \\ \hline 6207 & \phantom{-00}43449 \\ & - \phantom{00} 43449 \\ \hline & \phantom{000000}0 \\ \end{array} \] \[ \color{green}\sqrt{9653449} = 3107 \] \[ \begin{aligned} \sqrt{64432729} - \sqrt{9653449} &= 8027 - 3107 \\ &= 4920 \end{aligned} \]
Answer \( \color{red} \sqrt{64432729} - \sqrt{9653449} = 4920 \)