SIMPLIFYING :
[tex]1) \: \frac{6 + \sqrt{9} }{2(3)} \\ = \frac{6 + 3}{6} \\ = \frac{9}{6} = 1 \frac{1}{2} [/tex]
[tex]2) \: \frac{6 - \sqrt{9} }{2(3)} \\ = \frac{6 - 3}{6} \\ = \frac{3}{6} = \frac{1}{2} [/tex]
[tex]3) \: \frac{ - 6 + \sqrt{18} }{2(2)} \\ = \frac{ - 6 + 3\sqrt{2} }{4} \\ = \frac{3 \sqrt{2} }{4} = - 1 \frac{1}{2} [/tex]
[tex]4) \frac{ - 9 - \sqrt{24} }{2(2)} \\ = \frac{ - 9 - 2\sqrt{6} }{4} \\ = - 2\frac{1}{4} = \frac{ \sqrt{6} }{2} [/tex]
[tex]5) \: \frac{ - 8 + \sqrt{64 - 28} }{2( - 3)} \\ = \frac{ - 8 + \sqrt{36} }{ - 6} \\ = \frac{ - 8 + 6}{ - 6} = \frac{ - 2}{ - 6} \\ = \frac{1}{3} [/tex]
[tex]6) \: \frac{ - 6 - \sqrt{36 - 20} }{2} \\ = \frac{ - 6 - \sqrt{16} }{2} \\ = \frac{ - 6 - 4}{2} = \frac{ - 10}{2} \\ = - 5[/tex]
[tex]7) \frac{5 + \sqrt{25 + 100} }{2(4)} \\ = \frac{5 + \sqrt{125} }{8} \\ = \frac{5 + \sqrt{ {5}^{2} } \times \sqrt{5} }{8} \\ = \frac{5 + 5\sqrt{5} }{8} [/tex]
[tex]8) \frac{ - 10 + \sqrt{ {10}^{2}}- 52 }{2(3)} \\ = \frac{ - 10 \sqrt{100 - 52} }{6} \\ = \frac{ - 10 + \sqrt{48} }{6} \\ = \frac{ - 10 + \sqrt{ {4}^{2} \sqrt{3} } }{6} \\ = \frac{2( - 5 + 2 \sqrt{3)} }{6} \\ = \frac{ - 5 + 2 \sqrt{3} }{6} [/tex]
[tex]9) \frac{ - 4 - \sqrt{16 + 16} }{2(4)} \\ = \frac{ - 4 - 32}{8} \\ = \frac{ - 4 - \sqrt{ {4}^{2}} \times \sqrt 2 }{8} \\ = \frac{ - 4 \sqrt{4 \sqrt{2} } }{8} \\ = \frac{ - 4 \times (1 + \sqrt{2} }{8} \\ = - \frac{1 \sqrt{2} }{2} [/tex]
[tex]10) \frac{ - 5 + \sqrt{25 - 4 \times 3 \times 2} }{2(3)} \\ = \frac{ - 5 + \sqrt{25 - 24} }{6} \\ = \frac{ - 5 + \sqrt{1} }{6} \\ = \frac{ - 4}{6} \\ = - \frac{2}{3} [/tex]
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Answers & Comments
SIMPLIFYING :
[tex]1) \: \frac{6 + \sqrt{9} }{2(3)} \\ = \frac{6 + 3}{6} \\ = \frac{9}{6} = 1 \frac{1}{2} [/tex]
[tex]2) \: \frac{6 - \sqrt{9} }{2(3)} \\ = \frac{6 - 3}{6} \\ = \frac{3}{6} = \frac{1}{2} [/tex]
[tex]3) \: \frac{ - 6 + \sqrt{18} }{2(2)} \\ = \frac{ - 6 + 3\sqrt{2} }{4} \\ = \frac{3 \sqrt{2} }{4} = - 1 \frac{1}{2} [/tex]
[tex]4) \frac{ - 9 - \sqrt{24} }{2(2)} \\ = \frac{ - 9 - 2\sqrt{6} }{4} \\ = - 2\frac{1}{4} = \frac{ \sqrt{6} }{2} [/tex]
[tex]5) \: \frac{ - 8 + \sqrt{64 - 28} }{2( - 3)} \\ = \frac{ - 8 + \sqrt{36} }{ - 6} \\ = \frac{ - 8 + 6}{ - 6} = \frac{ - 2}{ - 6} \\ = \frac{1}{3} [/tex]
[tex]6) \: \frac{ - 6 - \sqrt{36 - 20} }{2} \\ = \frac{ - 6 - \sqrt{16} }{2} \\ = \frac{ - 6 - 4}{2} = \frac{ - 10}{2} \\ = - 5[/tex]
[tex]7) \frac{5 + \sqrt{25 + 100} }{2(4)} \\ = \frac{5 + \sqrt{125} }{8} \\ = \frac{5 + \sqrt{ {5}^{2} } \times \sqrt{5} }{8} \\ = \frac{5 + 5\sqrt{5} }{8} [/tex]
[tex]8) \frac{ - 10 + \sqrt{ {10}^{2}}- 52 }{2(3)} \\ = \frac{ - 10 \sqrt{100 - 52} }{6} \\ = \frac{ - 10 + \sqrt{48} }{6} \\ = \frac{ - 10 + \sqrt{ {4}^{2} \sqrt{3} } }{6} \\ = \frac{2( - 5 + 2 \sqrt{3)} }{6} \\ = \frac{ - 5 + 2 \sqrt{3} }{6} [/tex]
[tex]9) \frac{ - 4 - \sqrt{16 + 16} }{2(4)} \\ = \frac{ - 4 - 32}{8} \\ = \frac{ - 4 - \sqrt{ {4}^{2}} \times \sqrt 2 }{8} \\ = \frac{ - 4 \sqrt{4 \sqrt{2} } }{8} \\ = \frac{ - 4 \times (1 + \sqrt{2} }{8} \\ = - \frac{1 \sqrt{2} }{2} [/tex]
[tex]10) \frac{ - 5 + \sqrt{25 - 4 \times 3 \times 2} }{2(3)} \\ = \frac{ - 5 + \sqrt{25 - 24} }{6} \\ = \frac{ - 5 + \sqrt{1} }{6} \\ = \frac{ - 4}{6} \\ = - \frac{2}{3} [/tex]
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