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1)
![frac{1-2i}{1-3i} -(1-2i)(1-3i)+ i^{26}- i^{59} = \
= frac{(1-2i)(1+3i)}{(1-3i)(1+3i)} -(1-2i-3i+6 i^{2} )+i^{(24+2)}-i^{(56+3)}= \
= frac{1+3i-2i-6i^{2}}{1+3}-(1-5i-6)+ i^{2} - i^{3}= \
= frac{7+i}{4} +5+5i-1+i= \
= frac{7}{4} +5-1+( frac{1}{4}+5+1)i=5.75+6.25i](https://tex.z-dn.net/?f=+frac%7B1-2i%7D%7B1-3i%7D+-%281-2i%29%281-3i%29%2B+i%5E%7B26%7D-+i%5E%7B59%7D+%3D+%5C+%0A%3D+frac%7B%281-2i%29%281%2B3i%29%7D%7B%281-3i%29%281%2B3i%29%7D+-%281-2i-3i%2B6+i%5E%7B2%7D++%29%2Bi%5E%7B%2824%2B2%29%7D-i%5E%7B%2856%2B3%29%7D%3D+%5C+%0A%3D+frac%7B1%2B3i-2i-6i%5E%7B2%7D%7D%7B1%2B3%7D-%281-5i-6%29%2B+i%5E%7B2%7D+-+i%5E%7B3%7D%3D+%5C+%0A%3D+frac%7B7%2Bi%7D%7B4%7D+++%2B5%2B5i-1%2Bi%3D+%5C+%0A%3D+frac%7B7%7D%7B4%7D+%2B5-1%2B%28+frac%7B1%7D%7B4%7D%2B5%2B1%29i%3D5.75%2B6.25i+ "frac{1-2i}{1-3i} -(1-2i)(1-3i)+ i^{26}- i^{59} = \
= frac{(1-2i)(1+3i)}{(1-3i)(1+3i)} -(1-2i-3i+6 i^{2} )+i^{(24+2)}-i^{(56+3)}= \
= frac{1+3i-2i-6i^{2}}{1+3}-(1-5i-6)+ i^{2} - i^{3}= \
= frac{7+i}{4} +5+5i-1+i= \
= frac{7}{4} +5-1+( frac{1}{4}+5+1)i=5.75+6.25i")
2)
a)
![m+16n^{2}=( sqrt{m} +4ni)( sqrt{m}-4ni)=(4n+ sqrt{m}i)(4n- sqrt{m}i) \](https://tex.z-dn.net/?f=m%2B16n%5E%7B2%7D%3D%28+sqrt%7Bm%7D+%2B4ni%29%28+sqrt%7Bm%7D-4ni%29%3D%284n%2B+sqrt%7Bm%7Di%29%284n-+sqrt%7Bm%7Di%29+%5C+ "m+16n^{2}=( sqrt{m} +4ni)( sqrt{m}-4ni)=(4n+ sqrt{m}i)(4n- sqrt{m}i) \")
b)
3)
![x^{2} -2x+10=0 \
x_{12}= frac{2+- sqrt{2^{2}-4*10} }{2}=1+- sqrt{1-10} =1+- sqrt{-9}=1+- sqrt{3}i \ x_{1}=1- sqrt{3}i ; x_{2}=1+ sqrt{3}i \](https://tex.z-dn.net/?f=x%5E%7B2%7D+-2x%2B10%3D0+%5C+%0A+x_%7B12%7D%3D+frac%7B2%2B-+sqrt%7B2%5E%7B2%7D-4%2A10%7D+%7D%7B2%7D%3D1%2B-+sqrt%7B1-10%7D+++%3D1%2B-+sqrt%7B-9%7D%3D1%2B-+sqrt%7B3%7Di+%5C++x_%7B1%7D%3D1-+sqrt%7B3%7Di++%3B+x_%7B2%7D%3D1%2B+sqrt%7B3%7Di+%5C+++++ "x^{2} -2x+10=0 \
x_{12}= frac{2+- sqrt{2^{2}-4*10} }{2}=1+- sqrt{1-10} =1+- sqrt{-9}=1+- sqrt{3}i \ x_{1}=1- sqrt{3}i ; x_{2}=1+ sqrt{3}i \")
2)
a)
b)
3)
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