Explicație pas cu pas:
[tex]x = \left[ {2}^{ {30}^{2} } \times ( {2}^{6})^{100} \times 2 + ( {64}^{4})^{100} \div {2}^{899} \right]^{2} + {2}^{3007} = \\ [/tex]
[tex]= \left[ {2}^{900} \times {2}^{600} \times 2 + {( {2}^{6})}^{400} \div {2}^{899} \right]^{2} + {2}^{3007} \\ [/tex]
[tex]= \left({2}^{900 + 600 + 1} + {2}^{2400} \div {2}^{899} \right)^{2} + {2}^{3007} \\ [/tex]
[tex]= \left({2}^{1501} + {2}^{1501} \right)^{2} + {2}^{3007} = \left(2 \times {2}^{1501}\right)^{2} + {2}^{3007} \\ [/tex]
[tex]= \left({2}^{1502}\right)^{2} + {2}^{3007} = {2}^{3004} + {2}^{3007} \\ [/tex]
[tex]= {2}^{3004}(1 + {2}^{3})= {2}^{3004} \times 9 = {2}^{3004} \times {3}^{2} = {\left({2}^{1502} \times 3 \right)}^{2} \\ [/tex]
pentru y, am reinterpretat (așa cum ai scris, este imposibil să fie pătrat perfect, deoarece este un număr negativ):
[tex]y = 5 \times ( {3}^{2002} - {3}^{2001} - {9}^{1000}) = 5 \times ( {3}^{2}\times {3}^{2000} - 3\times {3}^{2000} - {({3}^{2} )}^{1000}) = 5 \times ( 9\times {3}^{2000} - 3 \times {3}^{2000} - {3}^{2000}) = 5 \times ( 5 \times {3}^{2000}) = {5}^{2} \times {3}^{2000} = (5 \times {3}^{1000})^{2}[/tex]
