Question

$\huge \sf{simplify - }$
$1. \: \sqrt{ \frac{5}{12} }$
$2. \: \sqrt{1 \frac{46}{75}}$
$3. \: \sqrt{112} - \sqrt{63} + \frac{224}{ \sqrt{28} }$
$4. \: \frac{4 \sqrt{18} }{ \sqrt{12} } - \frac{8 \sqrt{75} }{ \sqrt{32} } + \frac{9 \sqrt{2} }{ \sqrt{3} }$
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Answer 1

$\\\\ \sf1. \: \sqrt{ \dfrac{5}{12} }$

Solution

$\sf~~~~~~~~~~\implies \sqrt{ \dfrac{5}{12} }$

$\sf~~~~~~~~~~\implies \sqrt{ \dfrac{5}{2 \times 2 \times 3} }$

$\sf~~~~~~~~~~\implies \sqrt{ \dfrac{5}{ {2}^{2} \times 3} }$

$\sf~~~~~~~~~~\implies \dfrac{ \sqrt{5} }{ 2 \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{ \sqrt{5} }{ 2 \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{ \sqrt{15} }{ 2 \sqrt{3} \times \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{ \sqrt{15} }{ 2 \times 3 }$

$\sf~~~~~~~~~~\implies \dfrac{ \sqrt{15} }{ 6 }$

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$\\\bf \: \sqrt{ \dfrac{5}{12} } = \dfrac{ \sqrt{15} }{ 6 }$

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$\\ \sf2. \: \sqrt{1 \dfrac{46}{75}}$

Solution

$\sf~~~~~~~~~~\implies \sqrt{1 \dfrac{46}{75}}$

$\sf~~~~~~~~~~\implies \sqrt{ \dfrac{121}{75}}$

$\sf~~~~~~~~~~\implies \sqrt{ \dfrac{ {11}^{2} }{ {5}^{2} \times 3 }}$

$\sf~~~~~~~~~~\implies \dfrac{11 }{ 5 \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{11 }{ 5 \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{11 \sqrt{3} }{ 5 \times \sqrt{3} \times \sqrt{3} }$

$\sf~~~~~~~~~~\implies \dfrac{11 \sqrt{3} }{ 5 \times 3 }$

$\sf~~~~~~~~~~\implies \dfrac{11 \sqrt{3} }{ 15 }$

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$\\\bf \: \sqrt{1 \dfrac{46}{75}} = \dfrac{11 \sqrt{3} }{ 15 }$

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$\\ \sf3. \: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} }$

Solution

${\sf~~~~~~~~~~\implies \: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } }$

${\sf~~~~~~~~~~\implies \: \sqrt{2 \times 2 \times 2 \times 2 \times 7} - \sqrt{3 \times 3 \times 7} + \dfrac{224}{ \sqrt{2 \times 2 \times 7} } }$

${\sf~~~~~~~~~~\implies \: \sqrt{ {2}^{4} \times 7} - \sqrt{ {3}^{2} \times 7} + \dfrac{224}{ \sqrt{ {2}^{2} \times 7} } }$

${\sf~~~~~~~~~~\implies \: 4\sqrt{ 7} - 3\sqrt{ 7} + \dfrac{224}{ 2\sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \sqrt{ 7} + \dfrac{112}{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \dfrac{ \sqrt{7} \times \sqrt{7} + 112}{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \dfrac{ 7 + 112}{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \dfrac{ 119}{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \dfrac{ 17 \times 7}{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: \dfrac{ 17 \times \sqrt{7} \times \sqrt{7} }{ \sqrt{ 7} } }$

${\sf~~~~~~~~~~\implies \: 17 \sqrt{7} }$

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$\\ \bf \: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } = 17 \sqrt{7}$

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$\sf4.~~~~ \: \dfrac{4 \sqrt{18} }{ \sqrt{12} } - \dfrac{8 \sqrt{75} }{ \sqrt{32} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} }$

Solution

${\sf~~~~~~~~~~\implies \dfrac{4 \sqrt{18} }{ \sqrt{12} } - \dfrac{8 \sqrt{75} }{ \sqrt{32} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{4 \sqrt{2 \times 3 \times 3} }{ \sqrt{2 \times 2 \times 3} } - \dfrac{8 \sqrt{5 \times 5 \times 3} }{ \sqrt{2 \times 2 \times 2 \times 2 \times 2} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{4 \sqrt{2 \times {3}^{2} } }{ \sqrt{ {2}^{2} \times 3} } - \dfrac{8 \sqrt{ {5}^{2} \times 3} }{ \sqrt{ {4}^{2} \times 2} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{2 \times 2 \times 3 \sqrt{2 } }{ 2\sqrt{ 3} } - \dfrac{4 \times 2 \times 5 \sqrt{ 3} }{ 4\sqrt{ 2} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{ 6 \sqrt{2 } }{ \sqrt{ 3} } - \dfrac{10 \sqrt{ 3} }{ \sqrt{ 2} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{ 6 \sqrt{2 } }{ \sqrt{ 3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} } - \dfrac{10 \sqrt{ 3} }{ \sqrt{ 2} } \times \dfrac{ \sqrt{2} }{ \sqrt{2} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } \times \dfrac{ \sqrt{3} }{ \sqrt{3} } }$

${\sf~~~~~~~~~~\implies \dfrac{ 6 \sqrt{6 } }{ 3 } - \dfrac{10 \sqrt{ 6} }{ 2 } + \dfrac{9 \sqrt{6} }{ 3 } }$

${\sf~~~~~~~~~~\implies \dfrac{2( 6 \sqrt{6 }) - 3(10 \sqrt{6}) + 2(9 \sqrt{6} ) }{ 6 } }$

${\sf~~~~~~~~~~\implies \dfrac{12 \sqrt{6 } - 30 \sqrt{6} + 18 \sqrt{6} }{ 6 } }$

${\sf~~~~~~~~~~\implies \dfrac{ \cancel{30 \sqrt{6 }} - \cancel{3 0 \sqrt{6}} }{ 6 } }$

${\sf~~~~~~~~~~\implies \dfrac{ 0 }{ 6 } }$

${\sf~~~~~~~~~~\implies 0 }$

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${\bf \dfrac{4 \sqrt{18} }{ \sqrt{12} } - \dfrac{8 \sqrt{75} }{ \sqrt{32} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } = 0 }$

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$\\\bf \: \sqrt{ \dfrac{5}{12} } = \dfrac{ \sqrt{15} }{ 6 }$

$\bf \: \sqrt{1 \dfrac{46}{75}} = \dfrac{11 \sqrt{3} }{ 15 }$

$\bf \: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } = 17 \sqrt{7}$

${\bf \dfrac{4 \sqrt{18} }{ \sqrt{12} } - \dfrac{8 \sqrt{75} }{ \sqrt{32} } + \dfrac{9 \sqrt{2} }{ \sqrt{3} } = 0 }$

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Answer 2

$\bf{ 1 \sqrt{ \frac{5}{12} } \\ = \frac{ \sqrt{5}{ \sqrt{12} }$

$\bf{ = \frac{ \sqrt{5} }{ \sqrt{4 \times 3}} }$

$\bf{= \frac{ \sqrt{5} }{2 \sqrt{3} }}$

$\bf{2 \sqrt{ 1\frac{46}{75} } \\ = \sqrt{ \frac{121}{75} } }$

$\bf{ = \sqrt{ \frac{ {11}^{2} }{ {5}^{2} \times 3} } }$

$\bf{ = \frac{11}{5 \sqrt{3} }$

$\bf{= \frac{11}{5 \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } }$

$\bf{ = \frac{11 \sqrt{3} }{5 \times \sqrt{3} \times \sqrt{3} } }$

$\bf{= \frac{11 \sqrt{3} }{5 \times 3} }$

$\bf{= \frac{11 \sqrt{3} }{15} }$

$\bf{3 \sqrt{112} - 63 + \frac{224}{ \sqrt{28} } }$

$\bf{ = \sqrt{112} - \sqrt{63} + \frac{224}{ \sqrt{28} } }$

$\bf{ = \sqrt{2 \times 2 \times 2 \times 2 \times 7} - \sqrt{3 \times 3 \times 7} }$

$\bf{ = \sqrt{ {2}^{4} \times 7} - \sqrt{ {3}^{2} \times 7 } + \frac{224}{ {2}^{2} \times 7 }}$

$\bf{ = 4 \sqrt{7} - 3 \sqrt{7} + \frac{224}{2 \sqrt{7} } \\ = \sqrt{7} + \frac{112}{7} }$

$\bf{= \frac{ \sqrt{7} \times \sqrt{7} + 112}{ \sqrt{7} } \\ = \frac{7 + 112}{ \sqrt{7} } }$

$\bf{ = \frac{119}{ \sqrt{7} } }$

$\bf{= \frac{17 \times 7}{ \sqrt{7} } }$

$\bf{= \frac{17 \times \sqrt{7} \times \sqrt{7} }{ \sqrt{7} }}$

$\bf{= 17 \sqrt{7} }$

$\bf{4 \frac{4 \sqrt{8} }{ \sqrt{12} } - \frac{8 \sqrt{75} }{ \sqrt{32} } + \frac{9 \sqrt{2} }{3} }$

$\bf{ = \frac{4 \sqrt{2 \times 3 \times 3} }{ \sqrt{2 \times 2 \times 3} } - \frac{8 \sqrt{5 \times 5 \times 3} }{ \sqrt{2 \times 2 \times 2 \times 2 \times 2} } + \frac{9 \sqrt{2} }{ \sqrt{3} } }$

$\bf{ = \frac{4 \sqrt{2 \times {3}^{2} } }{ \sqrt{ {2}^{2} } \times 3 } - \frac{8 \sqrt{ {5}^{2} \times 3} }{ {4}^{2} \times 2 } + \frac{9 \sqrt{2} }{ \sqrt{3} } }$

$\bf{= \frac{2 \times 2 \times 3 \sqrt{2} }{2 \sqrt{3} } - \frac{4 \times 2 \times 5 \sqrt{3} }{4 \sqrt{2} } + \frac{9 \sqrt{3} }{ \sqrt{3} } }$

$\bf{= \frac{6 \sqrt{2} }{ \sqrt{3} } - \frac{10 \sqrt{3} }{ \sqrt{2} } + \frac{9 \sqrt{2} }{ \sqrt{3 } } }$

$\bf{ = \frac{6 \sqrt{2} }{ \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } - \frac{10 \sqrt{3} }{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } + \frac{9 \sqrt{2} }{ \sqrt{3} } \times \frac{ \sqrt{3} }{ \sqrt{3} } }$

$\bf{= \frac{6 \sqrt{6} }{3} - \frac{10 \sqrt{6} }{2} + \frac{9 \sqrt{6} }{3}}$

$\bf{= \frac{2(6 \sqrt{6} ) - 3(10 \sqrt{6} ) + 2(9 \sqrt{6}) }{6} }$

$\bf{= \frac{12 \sqrt{6} - 30 \sqrt{6} + 18 \sqrt{6} }{6} }$

$\bf{ = \frac{30 \sqrt{6} - 30 \sqrt{6} }{6} }$

$\bf{ = \frac{0}{6} = 0}$

#NAWABZAADI