Questions :1. Find the Square root of 1936 by prime factorization method. 2. Find the Square root of 4356 by prime factorization method. 3. Express (23)^2 as the Sum of two consecutive integers. About the author Ruby
Solution – 1 [tex] \sf \: First \: of \: all \: We \: have \: to \: find \: Prime \: factors \: of \: 1936.[/tex] [tex]\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:1936\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:968\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:484\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:242\:\:\:}}} \\{\underline{\sf{11}}}&{\underline{\sf{\:\:121\:\:\:}}} \\ {\underline{\sf{11}}}&{\underline{\sf{\:\:11\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}[/tex] [tex] \sf \: Now \: 1936 \: = \: { \sf \: \underline{2 \: \times \: 2} \: \times \: \underline {2 \: \times \: 2}} \: \times \: { \underline {11 \: \times \: 11}}[/tex] [tex] \sf \: \sqrt{1936} \: = \: \sqrt{2 \times 2 \times 2 \times 2 \times 11 \times 11} [/tex] [tex] \sf \therefore \: \sqrt{1936} \: = \: \sqrt{2 \times 2 \times 11 } [/tex] [tex] \sf \: \sqrt{1936} \: = 44[/tex] [tex]\underline{\boxed {\mathrm {\red{ \bigstar \: Thus \: 44, \: is \: the \: Square \: root \: of \: 1936 \: \bigstar} }}}[/tex] _____________________________________ Solution – 2 [tex] \sf \: First \: of \: all \: We \: have \: to \: find \: Prime \: factors \: of \: 4356.[/tex] [tex]\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:4356\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:2178\:\:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:1089\: \:\:}}\\{\underline{\sf{3}}}&{\underline{\sf{\:\:363\:\:\:}}} \\{\underline{\sf{11}}}&{\underline{\sf{\:\:121\:\:\:}}} \\ {\underline{\sf{11}}}&{\underline{\sf{\:\:11\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}[/tex] [tex] \sf \: So, \: 4356 \: = \: 2 \times 2 \times 3 \times 3 \times 11 \times 11[/tex] [tex] \sf \: Now, \: we \: will \: take \: one \: factor \: from \: each \: pair.[/tex] [tex] \sf \: \therefore \: \sqrt{4356} \: = \: \sqrt{ \underline{2 \times 2 }\times \underline{3\times 3}\times \underline{11 \times 11}}[/tex] [tex] \sf \: \sqrt{4356} \: = 2 \times 3\times 11 [/tex] [tex] \sf \: \sqrt{4356} \: = 66[/tex] [tex]\underline{\boxed {\mathrm {\red{ \bigstar \: Thus \: 66, \: is \: the \: Square \: root \: of \: 4356 \: \bigstar} }}}[/tex] _____________________________________ Solution – 3 [tex] \sf \: (23)^{2} \: = \: \dfrac{(23)^{2} \: + 1}{2} \: + \: \dfrac{(23)^{2} \: + 1}{2}[/tex] [tex] \sf \: = \: \dfrac{ \: 529 + 1}{2} \: + \: \dfrac{ 529 \: – 1}{2}[/tex] [tex] \sf \: = \: \dfrac{ \: 530}{2} \: + \: \dfrac{ 528}{2}[/tex] [tex]\underline{\boxed {\mathrm {\red{ \bigstar \: \: \sf \: = \: 256 \: + \: 264 \: = \: \bigstar } }}}[/tex] _____________________________________ Reply
Solution – 1
[tex] \sf \: First \: of \: all \: We \: have \: to \: find \: Prime \: factors \: of \: 1936.[/tex]
[tex]\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:1936\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:968\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:484\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:242\:\:\:}}} \\{\underline{\sf{11}}}&{\underline{\sf{\:\:121\:\:\:}}} \\ {\underline{\sf{11}}}&{\underline{\sf{\:\:11\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex] \sf \: Now \: 1936 \: = \: { \sf \: \underline{2 \: \times \: 2} \: \times \: \underline {2 \: \times \: 2}} \: \times \: { \underline {11 \: \times \: 11}}[/tex]
[tex] \sf \: \sqrt{1936} \: = \: \sqrt{2 \times 2 \times 2 \times 2 \times 11 \times 11} [/tex]
[tex] \sf \therefore \: \sqrt{1936} \: = \: \sqrt{2 \times 2 \times 11 } [/tex]
[tex] \sf \: \sqrt{1936} \: = 44[/tex]
[tex]\underline{\boxed {\mathrm {\red{ \bigstar \: Thus \: 44, \: is \: the \: Square \: root \: of \: 1936 \: \bigstar} }}}[/tex]
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Solution – 2
[tex] \sf \: First \: of \: all \: We \: have \: to \: find \: Prime \: factors \: of \: 4356.[/tex]
[tex]\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:4356\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:2178\:\:\:}} \\\underline{\sf{3}}&\underline{\sf{\:\:1089\: \:\:}}\\{\underline{\sf{3}}}&{\underline{\sf{\:\:363\:\:\:}}} \\{\underline{\sf{11}}}&{\underline{\sf{\:\:121\:\:\:}}} \\ {\underline{\sf{11}}}&{\underline{\sf{\:\:11\:\:\:}}} \\ \underline{\sf{}}&{\sf{\:\:1\:\:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex] \sf \: So, \: 4356 \: = \: 2 \times 2 \times 3 \times 3 \times 11 \times 11[/tex]
[tex] \sf \: Now, \: we \: will \: take \: one \: factor \: from \: each \: pair.[/tex]
[tex] \sf \: \therefore \: \sqrt{4356} \: = \: \sqrt{ \underline{2 \times 2 }\times \underline{3\times 3}\times \underline{11 \times 11}}[/tex]
[tex] \sf \: \sqrt{4356} \: = 2 \times 3\times 11 [/tex]
[tex] \sf \: \sqrt{4356} \: = 66[/tex]
[tex]\underline{\boxed {\mathrm {\red{ \bigstar \: Thus \: 66, \: is \: the \: Square \: root \: of \: 4356 \: \bigstar} }}}[/tex]
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Solution – 3
[tex] \sf \: (23)^{2} \: = \: \dfrac{(23)^{2} \: + 1}{2} \: + \: \dfrac{(23)^{2} \: + 1}{2}[/tex]
[tex] \sf \: = \: \dfrac{ \: 529 + 1}{2} \: + \: \dfrac{ 529 \: – 1}{2}[/tex]
[tex] \sf \: = \: \dfrac{ \: 530}{2} \: + \: \dfrac{ 528}{2}[/tex]
[tex]\underline{\boxed {\mathrm {\red{ \bigstar \: \: \sf \: = \: 256 \: + \: 264 \: = \: \bigstar } }}}[/tex]
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