Q1. Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication
$$(i) \quad 25 \quad (ii) \quad 37 \quad (iii) \quad 54 \quad (iv) \quad 71 \quad (v) 96 $$Sol. (i) 25
Here a=2, b=5
$$ \text {Step: 1 Make 3 columns and write the values of } a^{2}, 2 \times a \times b \text {, and } b^{2} \text { in these columns. } $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4 & 20 & 25 \\ \hline \end{array}$$ \text {Step: 2 Underline the unit digit of } b^{2} \text {(in Column III) and add its tens digit, if any, with } 2 \times a \times b $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4 & 20 + 2 & 2\underline{5} \\ \hline & 22 & \\ \hline \end{array}Step: 3 Underline the unit digit in Column II and add the number formed by
$$ \text {the tens and other digits if any, with } a^{2} \text { in Column I. } $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4+2 & 20 + 2 & 2\underline{5} \\ \hline 6 & 2\underline{2} & \\ \hline \end{array}Step: 4 Underline the number in Column I.
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4+2 & 20 + 2 & 2\underline{5} \\ \hline \underline{6} & 2\underline{2} & \\ \hline \end{array}Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.
$$ \text {In this case, we have: } 25^{2} = 625 $$$$ \text {Using Multiplication: } 25 \times 25 = 625 $$(ii) 37
Here a=3, b=7, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 9 & 42 & 4\underline{9} \\ \hline 9 & 42 + 4 & \\ \hline 9+4 & 4\underline{6} & \\ \hline \underline{13} & & \\ \hline \end{array}$$ \therefore 37^{2} = 1369 $$$$ \text {Using Multiplication: } 37 \times 37 = 1369 $$(iii) 54
Here a=5, b=4, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 25 & 40 & 1\underline{6} \\ \hline 25 & 40 + 1 & \\ \hline 25+4 & 4\underline{1} & \\ \hline \underline{29} & & \\ \hline \end{array}$$ \therefore 54^{2} = 2916 $$$$ \text {Using Multiplication: } 54 \times 54 = 2916 $$(iv) 71
Here a=7, b=1, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 49 & 14 & \underline{1} \\ \hline 49 & 14 + 0 & \\ \hline 49+1 & 1\underline{4} & \\ \hline \underline{50} & & \\ \hline \end{array}$$ \therefore 71^{2} = 5041 $$$$ \text {Using Multiplication: } 71 \times 71 = 5041 $$(v) 96
Here a=9, b=6, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 81 & 108 & 3\underline{6} \\ \hline 81 & 108 + 3 & \\ \hline 81+11 & 11\underline{1} & \\ \hline \underline{92} & & \\ \hline \end{array}$$ \therefore 96^{2} = 9216 $$$$ \text {Using Multiplication: } 96 \times 96 = 9216 $$Q2. Find the squares of the following numbers using diagonal method:
$$ (i) \quad 98 \quad (ii) 273 \quad (iii) \quad 348 $$ $$ (iv) \quad 295 \quad (v) \quad 17 $$\begin{array}{l} \text {Need to add solution } \\ \end{array}Q3. Find the squares of the following numbers:
(i) 127
(ii) 503
(iii) 451
(iv) 862
(v) 265
Sol.
$$ (i) \quad 127^{2} = 127 \times 127 = 16129 $$ $$ (ii) \quad 503^{2} = 503 \times 503 = 253009 $$ $$ (iii) \quad 451^{2} = 451 \times 451 = 203401 $$ $$ (iv) \quad 862^{2} = 862 \times 862 = 743044 $$ $$ (v) \quad 265^{2} = 265 \times 265 = 70225 $$Q4. Find the squares of the following numbers:
(i) 425
(ii) 575
(iii) 405
(iv) 205
(v) 95
(vi) 745
(vii) 512
(viil) 995
Sol. $$ (i) 425^{2} = 425 \times 425 = 180625 $$Alternatively, we can find square by using the rule which says
The square of a number abc…5 (i.e a number having 5 at unit’s place is obtained by affixing 25 to the right of the number (n)(n+1), where n = abc…)
In 425, n = 42
$$ \therefore n(n+1) = 42 \times 43 = 1806 $$ $$ \Rightarrow \quad 425^{2} = 180625 $$$$ (ii) \quad 575^{2} = 575 \times 575 = 330625 $$ $$ (iii) \quad 405^{2} = 405 \times 405 = 164025 $$ $$ (iv) \quad 205^{2} = 205 \times 205 = 42025 $$ $$ (v) \quad 95^{2} = 95 \times 95 = 9025 $$ $$ (vi) \quad 745^{2} = 745 \times 745 = 555025 $$ $$ (vii) \quad 512^{2} = 512 \times 512 = 262144 $$Alternatively, we can find square by using the rule which says
$$ (5ab)^{2} = (250 + ab)1000 + (ab)^{2} $$ $$ \text {Here in 512, a = 1, b = 2 } $$ $$ \therefore \quad 512^{2}=(250+12) 1000+(12)^{2}=262000+144=262144 $$$$ \text {(viil) } 995^{2} = 995 \times 995 = 990025 $$
$$(i) \quad 25 \quad (ii) \quad 37 \quad (iii) \quad 54 \quad (iv) \quad 71 \quad (v) 96 $$Sol. (i) 25
Here a=2, b=5
$$ \text {Step: 1 Make 3 columns and write the values of } a^{2}, 2 \times a \times b \text {, and } b^{2} \text { in these columns. } $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4 & 20 & 25 \\ \hline \end{array}$$ \text {Step: 2 Underline the unit digit of } b^{2} \text {(in Column III) and add its tens digit, if any, with } 2 \times a \times b $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4 & 20 + 2 & 2\underline{5} \\ \hline & 22 & \\ \hline \end{array}Step: 3 Underline the unit digit in Column II and add the number formed by
$$ \text {the tens and other digits if any, with } a^{2} \text { in Column I. } $$\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4+2 & 20 + 2 & 2\underline{5} \\ \hline 6 & 2\underline{2} & \\ \hline \end{array}Step: 4 Underline the number in Column I.
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 4+2 & 20 + 2 & 2\underline{5} \\ \hline \underline{6} & 2\underline{2} & \\ \hline \end{array}Step: 5 write the underlined digits at the bottom of each column to obtain the square of the given number.
$$ \text {In this case, we have: } 25^{2} = 625 $$$$ \text {Using Multiplication: } 25 \times 25 = 625 $$(ii) 37
Here a=3, b=7, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 9 & 42 & 4\underline{9} \\ \hline 9 & 42 + 4 & \\ \hline 9+4 & 4\underline{6} & \\ \hline \underline{13} & & \\ \hline \end{array}$$ \therefore 37^{2} = 1369 $$$$ \text {Using Multiplication: } 37 \times 37 = 1369 $$(iii) 54
Here a=5, b=4, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 25 & 40 & 1\underline{6} \\ \hline 25 & 40 + 1 & \\ \hline 25+4 & 4\underline{1} & \\ \hline \underline{29} & & \\ \hline \end{array}$$ \therefore 54^{2} = 2916 $$$$ \text {Using Multiplication: } 54 \times 54 = 2916 $$(iv) 71
Here a=7, b=1, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 49 & 14 & \underline{1} \\ \hline 49 & 14 + 0 & \\ \hline 49+1 & 1\underline{4} & \\ \hline \underline{50} & & \\ \hline \end{array}$$ \therefore 71^{2} = 5041 $$$$ \text {Using Multiplication: } 71 \times 71 = 5041 $$(v) 96
Here a=9, b=6, We have
\begin{array}{|l|l|l|} \hline \text {Column I } & \text {Column II } & \text {Column III} \\ \hline a^{2} & 2 \times a \times b & b^{2} \\ \hline 81 & 108 & 3\underline{6} \\ \hline 81 & 108 + 3 & \\ \hline 81+11 & 11\underline{1} & \\ \hline \underline{92} & & \\ \hline \end{array}$$ \therefore 96^{2} = 9216 $$$$ \text {Using Multiplication: } 96 \times 96 = 9216 $$Q2. Find the squares of the following numbers using diagonal method:
$$ (i) \quad 98 \quad (ii) 273 \quad (iii) \quad 348 $$ $$ (iv) \quad 295 \quad (v) \quad 17 $$\begin{array}{l} \text {Need to add solution } \\ \end{array}Q3. Find the squares of the following numbers:
(i) 127
(ii) 503
(iii) 451
(iv) 862
(v) 265
Sol.
$$ (i) \quad 127^{2} = 127 \times 127 = 16129 $$ $$ (ii) \quad 503^{2} = 503 \times 503 = 253009 $$ $$ (iii) \quad 451^{2} = 451 \times 451 = 203401 $$ $$ (iv) \quad 862^{2} = 862 \times 862 = 743044 $$ $$ (v) \quad 265^{2} = 265 \times 265 = 70225 $$Q4. Find the squares of the following numbers:
(i) 425
(ii) 575
(iii) 405
(iv) 205
(v) 95
(vi) 745
(vii) 512
(viil) 995
Sol. $$ (i) 425^{2} = 425 \times 425 = 180625 $$Alternatively, we can find square by using the rule which says
The square of a number abc…5 (i.e a number having 5 at unit’s place is obtained by affixing 25 to the right of the number (n)(n+1), where n = abc…)
In 425, n = 42
$$ \therefore n(n+1) = 42 \times 43 = 1806 $$ $$ \Rightarrow \quad 425^{2} = 180625 $$$$ (ii) \quad 575^{2} = 575 \times 575 = 330625 $$ $$ (iii) \quad 405^{2} = 405 \times 405 = 164025 $$ $$ (iv) \quad 205^{2} = 205 \times 205 = 42025 $$ $$ (v) \quad 95^{2} = 95 \times 95 = 9025 $$ $$ (vi) \quad 745^{2} = 745 \times 745 = 555025 $$ $$ (vii) \quad 512^{2} = 512 \times 512 = 262144 $$Alternatively, we can find square by using the rule which says
$$ (5ab)^{2} = (250 + ab)1000 + (ab)^{2} $$ $$ \text {Here in 512, a = 1, b = 2 } $$ $$ \therefore \quad 512^{2}=(250+12) 1000+(12)^{2}=262000+144=262144 $$$$ \text {(viil) } 995^{2} = 995 \times 995 = 990025 $$