[tex]\large\underline{\sf{Solution-}}[/tex]
Given number is 243.
Now, we have to find by which least number should 243 be multiplied to make it a perfect square.
So, in order to find by which least number should 243 be multiplied to make it a perfect square, we use method of prime factorization.
So, using method of prime factorization, we have
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:243 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:81\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex]\implies\sf \: 243 = 3\times 3 \times 3 \times 3 \times 3 \\ [/tex]
So, from this we concluded that in the prime factorization of 243, one 3 not appear in pairs.
So, least number by which 243 should be multiplied to make it a perfect square is 3
Thus, Required number = 243 × 3 = 729
Now, Consider [tex]\sf \: \sqrt{729} \\ [/tex]
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:729 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:81\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex]\sf \: 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \\ [/tex]
[tex]\implies\sf \: \sqrt{729} = 3 \times 3 \times 3 \\ [/tex]
[tex]\implies\sf \: \sqrt{729} = 27 \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]
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Verified answer
[tex]\large\underline{\sf{Solution-}}[/tex]
Given number is 243.
Now, we have to find by which least number should 243 be multiplied to make it a perfect square.
So, in order to find by which least number should 243 be multiplied to make it a perfect square, we use method of prime factorization.
So, using method of prime factorization, we have
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:243 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:81\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex]\implies\sf \: 243 = 3\times 3 \times 3 \times 3 \times 3 \\ [/tex]
So, from this we concluded that in the prime factorization of 243, one 3 not appear in pairs.
So, least number by which 243 should be multiplied to make it a perfect square is 3
Thus, Required number = 243 × 3 = 729
Now, Consider [tex]\sf \: \sqrt{729} \\ [/tex]
So, using method of prime factorization, we have
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{3}}}&{\underline{\sf{\:\:729 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:243 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:81\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]
[tex]\sf \: 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \\ [/tex]
[tex]\implies\sf \: \sqrt{729} = 3 \times 3 \times 3 \\ [/tex]
[tex]\implies\sf \: \sqrt{729} = 27 \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional information
[tex]\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}[/tex]