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Saaderm
[tex]1)\ 255(2^8+1)(2^{16}+1)(2^{32}+1)=(256-1)(2^8+1)(2^{16}+1)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=(2^8-1)(2^8+1)(2^{16}+1)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=[(2^8)^2-1^2)(2^{16}+1)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=(2^{16}-1)(2^{16}+1)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=[(2^{16})^2-1^2)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=(2^{32}-1)(2^{32}+1)\\\\255(2^8+1)(2^{16}+1)(2^{32}+1)=[(2^{32})^2-1^2)\\\\\boxed{255(2^8+1)(2^{16}+1)(2^{32}+1)=2^{64}-1}[/tex]
[tex]2)\ (333333)^2+(444444)^2=(3\times111111)^2+(4\times111111)^2\\\\(333333)^2+(444444)^2=3^2\times111111^2+4^2\times111111^2\\\\(333333)^2+(444444)^2=9\times111111^2+16\times111111^2\\\\(333333)^2+(444444)^2=(9+16)\times111111^2\\\\(333333)^2+(444444)^2=25\times111111^2\\\\(333333)^2+(444444)^2=5^2\times111111^2\\\\(333333)^2+(444444)^2=(5\times111111)^2\\\\\boxed{(333333)^2+(444444)^2=555555^2}[/tex]
[tex]3)\ (499999)^2+999999=(500000-1)^2+(1000000-1)\\\\(499999)^2+999999=(5\times100000-1)^2+(10^6-1)\\\\(499999)^2+999999=(5\times10^5-1)^2+(10^6-1)\\\\(499999)^2+999999=25\times10^{10}-2\times5\times10^5+1+10^6-1\\\\(499999)^2+999999=25\times10^{10}-10\times10^5+1+10^6-1\\\\(499999)^2+999999=25\times10^{10}-10^{1+5}+10^6+1-1\\\\(499999)^2+999999=25\times10^{10}-10^{6}+10^6+1-1\\\\\boxed{(499999)^2+999999=25\times10^{10}}[/tex]
[tex]4)\ (999999)^2+(2000)^2=(1000000-1)^2+(2\times1000)^2\\\\(999999)^2+(2000)^2=(10^6-1)^2+(2\times10^3)^2\\\\(999999)^2+(2000)^2=10^{12}-2\times10^6+1+4\times10^6\\\\(999999)^2+(2000)^2=10^{12}+2\times10^6+1\\\\(999999)^2+(2000)^2=(10^{6}+1)^2\\\\(999999)^2+(2000)^2=(1000000+1)^2\\\\\boxed{(999999)^2+(2000)^2=(1000001)^2}[/tex]
