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a) \(85^2+75^2+65^2+55^2-45^2-35^2-25^2-15^2\)
\(=\left(85^2-15^2\right)+\left(75^2-25^2\right)+\left(65^2-35^2\right)+\left(55^2-45^2\right)\)
\(=\left(85-15\right)\left(85+15\right)+\left(75-25\right)\left(75+25\right)+\left(65-35\right)\left(65+35\right)+\left(55-45\right)\left(55+45\right)\)
\(=70.100+50.100+30.100+10.100\)
\(=7000+5000+3000+1000\)
\(=16000\)
b) \(\frac{135^2+130.135+65^2}{135^2-65^2}\)
\(=\frac{135^2+2.60.135+65^2}{135^2-65^2}\)
\(=\frac{\left(135+65\right)^2}{\left(135-65\right)^2}\)
\(=\frac{200^2}{70^2}\) \(=\frac{200}{70}=\frac{20}{7}\)
Thank you very much!!!!
\(M=\dfrac{\left(a-b\right)^3-c^3+3ab\left(a-b\right)-3abc}{\left(a+b\right)^2+\left(b-c\right)^2+\left(c+a\right)^2}\)
\(=\dfrac{\left(a-b-c\right)\left(a^2-2ab+b^2+ac-bc+c^2+3ab\right)}{2a^2+2b^2+2c^2+2ab-2bc+2ac}\)
\(=\dfrac{\left(a-b-c\right)\cdot\left(a^2+b^2+c^2-ab-bc+ac\right)}{2\cdot\left(a^2+b^2+c^2+ab-bc+ac\right)}=\dfrac{2}{2}=1\)
\(S=\dfrac{\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc}{2a^2+2b^2+2c^2-2ab-2bc-2ac}\)
\(=\dfrac{\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)}{2a^2+2b^2+2c^2-2ab-2bc-2ac}\)
\(=\dfrac{3\cdot\left(2a^2+2b^2+2c^2-2ab-2bc-2ac\right)\cdot\dfrac{1}{2}}{2a^2+2b^2+2c^2-2ab-2bc-2ac}=\dfrac{3}{2}\)
Đặt A = \(\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(2.\)\(\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
2A = \(\left(3^{32}-1\right)\left(3^{32}+1\right)\)
2A = \(3^{64}-1\)
A = \(\dfrac{3^{64}-1}{2}\)
\(\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)
\(=\left(3-1\right)\left(3+1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right).\dfrac{1}{2}\)
\(=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right).\dfrac{1}{2}\)
\(=\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right).\dfrac{1}{2}\)
\(=\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right).\dfrac{1}{2}\)\(=\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{32}+1\right).\dfrac{1}{2}=\left(3^{32}-1\right)\left(3^{32}+1\right).\dfrac{1}{2}\)
\(=\left(3^{64}-1\right).\dfrac{1}{2}=\dfrac{3^{64}-1}{2}\)
\(S=-1^2+2^2-3^2+4^2-...+2016^2\)
\(=\left(2-1\right)\left(2+1\right)+\left(4-3\right)\left(4+3\right)+...+\left(2016-2015\right)\left(2016+2015\right)\)
\(=3+7+..+4031\)
\(=2033136\)
\(A=\left(4^2+1\right)\left(4^4+1\right)\left(4^8+1\right)\left(4^{16}+1\right)\left(4^{32}+1\right)-\frac{1}{15}\times4^{64}\)
\(15A=\left(4^2-1\right)\left(4^2+1\right)\left(4^4+1\right)\left(4^{16}+1\right)\left(4^{32}+1\right)-4^{64}\)
\(15A=\left(4^4-1\right)\left(4^4+1\right)\left(4^{16}+1\right)\left(4^{32}+1\right)-4^{64}\)
\(15A=\left(4^{16}-1\right)\left(4^{16}+1\right)\left(4^{32}+1\right)-4^{64}\)
\(15A=\left(4^{32}-1\right)\left(4^{32}+1\right)-4^{64}\left(4^{32}\right)\)
\(15A=4^{64}-1-4^{64}\)
\(A=-\frac{1}{15}\)