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Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

Given number is 720.

Now, we have to find by which smallest number should 720 be multiplied to make it a perfect square.

So, in order to find by which smallest number should 720 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{2}}}&{\underline{\sf{\:\:720 \:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:360 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:120 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:60 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:30\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]

[tex]\implies\sf \: 720 = 2 \times 3 \times 2 \times 2 \times 2 \times 3 \times 5 \\ [/tex]

So, from this we concluded that in the prime factorization of 720, 5 occur only once.

So, smallest number by which 720 should be multiplied to make it a perfect square is 5

Thus, Required number = 720 × 5 = 3600

Now, Consider [tex]\sf \: \sqrt{3600} \\ [/tex]

So, using method of prime factorization, we have

[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:3600 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:720 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:360 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:120 \:\:}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:60 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:30\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:15 \:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}\end{gathered}\end{gathered}[/tex]

[tex]\sf \: 3600 = 5 \times 2 \times 3 \times 2 \times 2 \times 2 \times 3 \times 5 \\ [/tex]

[tex]\sf \: 3600 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \\ [/tex]

[tex]\implies\sf \: \sqrt{3600} = 2 \times 2 \times 3 \times 5 \\ [/tex]

[tex]\implies\sf \: \sqrt{3600} = 60 \\ [/tex]

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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]