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1.
![a)6* sqrt{1 frac{7}{9} } -4=6 sqrt{ frac{16}{9} } -4=6* frac{4}{3} -4=2*4-4=8-4=4 a)6* sqrt{1 frac{7}{9} } -4=6 sqrt{ frac{16}{9} } -4=6* frac{4}{3} -4=2*4-4=8-4=4](https://tex.z-dn.net/?f=a%296%2A+sqrt%7B1+frac%7B7%7D%7B9%7D+%7D+-4%3D6+sqrt%7B+frac%7B16%7D%7B9%7D+%7D+-4%3D6%2A+frac%7B4%7D%7B3%7D+-4%3D2%2A4-4%3D8-4%3D4)
![b) sqrt{7,2} * sqrt{20} = sqrt{144} =12 b) sqrt{7,2} * sqrt{20} = sqrt{144} =12](https://tex.z-dn.net/?f=b%29+sqrt%7B7%2C2%7D+%2A+sqrt%7B20%7D+%3D+sqrt%7B144%7D+%3D12)
![c) frac{ sqrt{216} }{ sqrt{6} } = frac{6 sqrt{6} }{ sqrt{6} } =6 c) frac{ sqrt{216} }{ sqrt{6} } = frac{6 sqrt{6} }{ sqrt{6} } =6](https://tex.z-dn.net/?f=c%29+frac%7B+sqrt%7B216%7D+%7D%7B+sqrt%7B6%7D+%7D+%3D+frac%7B6+sqrt%7B6%7D+%7D%7B+sqrt%7B6%7D+%7D+%3D6)
![d) sqrt{5^4*3^2} =5^2*3=25*3=75 d) sqrt{5^4*3^2} =5^2*3=25*3=75](https://tex.z-dn.net/?f=d%29+sqrt%7B5%5E4%2A3%5E2%7D+%3D5%5E2%2A3%3D25%2A3%3D75)
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![a) 4sqrt{20} - sqrt{125} =4*2 sqrt{5} -5 sqrt{5} =8 sqrt{5} -5 sqrt{5} =3 sqrt{5} a) 4sqrt{20} - sqrt{125} =4*2 sqrt{5} -5 sqrt{5} =8 sqrt{5} -5 sqrt{5} =3 sqrt{5}](https://tex.z-dn.net/?f=a%29+4sqrt%7B20%7D+-+sqrt%7B125%7D+%3D4%2A2+sqrt%7B5%7D+-5+sqrt%7B5%7D+%3D8+sqrt%7B5%7D+-5+sqrt%7B5%7D+%3D3+sqrt%7B5%7D+)
![b)( 3 sqrt{6} + sqrt{12} )* sqrt{3} =(3 sqrt{6} +2 sqrt{3} ) *sqrt{3} =3 sqrt{18} +6=3*3 sqrt{2} +6= b)( 3 sqrt{6} + sqrt{12} )* sqrt{3} =(3 sqrt{6} +2 sqrt{3} ) *sqrt{3} =3 sqrt{18} +6=3*3 sqrt{2} +6=](https://tex.z-dn.net/?f=b%29%28+3+sqrt%7B6%7D+%2B+sqrt%7B12%7D+%29%2A+sqrt%7B3%7D+%3D%283+sqrt%7B6%7D+%2B2+sqrt%7B3%7D+%29+%2Asqrt%7B3%7D+%3D3+sqrt%7B18%7D+%2B6%3D3%2A3+sqrt%7B2%7D+%2B6%3D)
![=9 sqrt{2} +6 =9 sqrt{2} +6](https://tex.z-dn.net/?f=%3D9+sqrt%7B2%7D+%2B6)
![c) (5- sqrt{2} )^2=25-10 sqrt{2} +2=27-10 sqrt{2} c) (5- sqrt{2} )^2=25-10 sqrt{2} +2=27-10 sqrt{2}](https://tex.z-dn.net/?f=c%29+%285-+sqrt%7B2%7D+%29%5E2%3D25-10+sqrt%7B2%7D+%2B2%3D27-10+sqrt%7B2%7D+)
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![frac{1}{2} sqrt{12} = frac{1}{2}*2 sqrt{3} = sqrt{3} frac{1}{2} sqrt{12} = frac{1}{2}*2 sqrt{3} = sqrt{3}](https://tex.z-dn.net/?f=+frac%7B1%7D%7B2%7D+sqrt%7B12%7D+%3D+frac%7B1%7D%7B2%7D%2A2+sqrt%7B3%7D+%3D+sqrt%7B3%7D+)
![frac{1}{3} sqrt{45} = frac{1}{3} *3 sqrt{5} = sqrt{5} frac{1}{3} sqrt{45} = frac{1}{3} *3 sqrt{5} = sqrt{5}](https://tex.z-dn.net/?f=+frac%7B1%7D%7B3%7D+sqrt%7B45%7D+%3D+frac%7B1%7D%7B3%7D+%2A3+sqrt%7B5%7D+%3D+sqrt%7B5%7D+)
![sqrt{3} leq sqrt{5} sqrt{3} leq sqrt{5}](https://tex.z-dn.net/?f=+sqrt%7B3%7D+leq+sqrt%7B5%7D+)
4.
![a) frac{6- sqrt{6} }{ sqrt{18} - sqrt{3} } = frac{6- sqrt{6} }{3 sqrt{2} - sqrt{3} } = frac{(6- sqrt{6})(3 sqrt{2}+ sqrt{3} ) }{15} = frac{18 sqrt{2}+6 sqrt{3} -3 sqrt{12} - sqrt{18} }{15} = a) frac{6- sqrt{6} }{ sqrt{18} - sqrt{3} } = frac{6- sqrt{6} }{3 sqrt{2} - sqrt{3} } = frac{(6- sqrt{6})(3 sqrt{2}+ sqrt{3} ) }{15} = frac{18 sqrt{2}+6 sqrt{3} -3 sqrt{12} - sqrt{18} }{15} =](https://tex.z-dn.net/?f=a%29+frac%7B6-+sqrt%7B6%7D+%7D%7B+sqrt%7B18%7D+-+sqrt%7B3%7D+%7D+%3D+frac%7B6-+sqrt%7B6%7D+%7D%7B3+sqrt%7B2%7D+-+sqrt%7B3%7D+%7D+%3D+frac%7B%286-+sqrt%7B6%7D%29%283+sqrt%7B2%7D%2B+sqrt%7B3%7D+%29+%7D%7B15%7D+%3D+frac%7B18+sqrt%7B2%7D%2B6+sqrt%7B3%7D+-3+sqrt%7B12%7D+-+sqrt%7B18%7D+%7D%7B15%7D+%3D)
![= frac{18 sqrt{2}+6 sqrt{3}-6 sqrt{3} - sqrt{18} }{15} = frac{18 sqrt{2}-3 sqrt{2} }{15} = frac{15 sqrt{2} }{15} = sqrt{2} = frac{18 sqrt{2}+6 sqrt{3}-6 sqrt{3} - sqrt{18} }{15} = frac{18 sqrt{2}-3 sqrt{2} }{15} = frac{15 sqrt{2} }{15} = sqrt{2}](https://tex.z-dn.net/?f=%3D+frac%7B18+sqrt%7B2%7D%2B6+sqrt%7B3%7D-6+sqrt%7B3%7D+-+sqrt%7B18%7D+%7D%7B15%7D+%3D+frac%7B18+sqrt%7B2%7D-3+sqrt%7B2%7D%C2%A0+%7D%7B15%7D+%3D+frac%7B15+sqrt%7B2%7D+%7D%7B15%7D+%3D+sqrt%7B2%7D+)
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