[tex]\large\underline{\sf{Solution-8}}[/tex]
Given number is 252.
Now, we have to find by which smallest number should 252 be multiplied to make it a perfect square.
So, in order to find by which smallest number should 252 be multiplied to make it a perfect square, we use method of prime factorization.
So, using method of prime factorization, we have
[tex] \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:252\:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:126\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:63 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:21 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}[/tex]
[tex]\implies\sf \: 252 = 2 \times 2 \times 3 \times 3 \times 7 \\ [/tex]
So, from this we concluded that in the prime factorization of 252, 7 occur only once.
So, smallest number by which 252 should be multiplied to make it a perfect square is 7.
Thus, Required number = 252 × 7 = 1764
[tex]\large\underline{\sf{Solution-9(a)}}[/tex]
[tex]\sf\: 8x + 8(x + 1) + 8(x + 2) = 888 \\ [/tex]
[tex]\sf\: 8(x + x + 1 + x + 2) = 888 \\ [/tex]
[tex]\sf\: 3x + 3 = 111 \\ [/tex]
[tex]\sf\: 3x = 111 - 3 \\ [/tex]
[tex]\sf\: 3x = 108 \\ [/tex]
[tex]\implies\sf\:\boxed{\bf\:x = 36 \: } \\ [/tex]
[tex]\large\underline{\sf{Solution-9(b)}}[/tex]
[tex]\sf\: 2(2x + 2 + x) = 154 \\ [/tex]
[tex]\sf\: 2(3x + 2) = 154 \\ [/tex]
[tex]\sf\: 3x + 2 = \dfrac{154}{2} \\ [/tex]
[tex]\sf\: 3x + 2 = 77 \\ [/tex]
[tex]\sf\: 3x = 77 - 2 \\ [/tex]
[tex]\sf\: 3x = 75 \\ [/tex]
[tex]\implies\sf\:\boxed{\bf\:x = 25 \: } \\ [/tex]
[tex]\large\underline{\sf{Solution-10}}[/tex]
[tex]\sf\: \sqrt{4489} \\ [/tex]
[tex] \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{ \: \: \: \: 67 \: \: \: \: }}}\\ {\underline{\sf{7}}}& {\sf{4489}} \\{\sf{}}& \underline{\sf{36 \: \: \: \: }} \\ {\underline{\sf{127}}}& {\sf{889}} \\{\sf{}}& \underline{\sf{889}} \\ {\underline{\sf{}}}& {\sf{ \: \: 0 \: \: }} \end{array}[/tex]
Thus,
[tex]\implies\sf\:\boxed{\bf\: \sqrt{4489} = 67 \: } \\ [/tex]
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Verified answer
[tex]\large\underline{\sf{Solution-8}}[/tex]
Given number is 252.
Now, we have to find by which smallest number should 252 be multiplied to make it a perfect square.
So, in order to find by which smallest number should 252 be multiplied to make it a perfect square, we use method of prime factorization.
So, using method of prime factorization, we have
[tex] \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:252\:\:}}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:126\:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:63 \:\:}}\\ {\underline{\sf{3}}}& \underline{\sf{\:\:21 \:\:}} \\ {\underline{\sf{7}}}& \underline{\sf{\:\:7\:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}[/tex]
[tex]\implies\sf \: 252 = 2 \times 2 \times 3 \times 3 \times 7 \\ [/tex]
So, from this we concluded that in the prime factorization of 252, 7 occur only once.
So, smallest number by which 252 should be multiplied to make it a perfect square is 7.
Thus, Required number = 252 × 7 = 1764
[tex]\large\underline{\sf{Solution-9(a)}}[/tex]
[tex]\sf\: 8x + 8(x + 1) + 8(x + 2) = 888 \\ [/tex]
[tex]\sf\: 8(x + x + 1 + x + 2) = 888 \\ [/tex]
[tex]\sf\: 3x + 3 = 111 \\ [/tex]
[tex]\sf\: 3x = 111 - 3 \\ [/tex]
[tex]\sf\: 3x = 108 \\ [/tex]
[tex]\implies\sf\:\boxed{\bf\:x = 36 \: } \\ [/tex]
[tex]\large\underline{\sf{Solution-9(b)}}[/tex]
[tex]\sf\: 2(2x + 2 + x) = 154 \\ [/tex]
[tex]\sf\: 2(3x + 2) = 154 \\ [/tex]
[tex]\sf\: 3x + 2 = \dfrac{154}{2} \\ [/tex]
[tex]\sf\: 3x + 2 = 77 \\ [/tex]
[tex]\sf\: 3x = 77 - 2 \\ [/tex]
[tex]\sf\: 3x = 75 \\ [/tex]
[tex]\implies\sf\:\boxed{\bf\:x = 25 \: } \\ [/tex]
[tex]\large\underline{\sf{Solution-10}}[/tex]
[tex]\sf\: \sqrt{4489} \\ [/tex]
[tex] \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{ \: \: \: \: 67 \: \: \: \: }}}\\ {\underline{\sf{7}}}& {\sf{4489}} \\{\sf{}}& \underline{\sf{36 \: \: \: \: }} \\ {\underline{\sf{127}}}& {\sf{889}} \\{\sf{}}& \underline{\sf{889}} \\ {\underline{\sf{}}}& {\sf{ \: \: 0 \: \: }} \end{array}[/tex]
Thus,
[tex]\implies\sf\:\boxed{\bf\: \sqrt{4489} = 67 \: } \\ [/tex]