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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) \(\frac{135^2+130.135+65^2}{135^2-65^2}\)
\(=\frac{135^2+2.60.135+65^2}{135^2-65^2}\)
\(=\frac{\left(135+65\right)^2}{\left(135-65\right)^2}\)
\(=\frac{200^2}{70^2}\) \(=\frac{200}{70}=\frac{20}{7}\)
964 - 1 = (932 + 1)(932 - 1) = ... = (932 + 1)(916 + 1)(98 + 1)(94 + 1)(92 + 1)(9 + 1)(9 - 1) > (932 + 1)(916 + 1)(98 + 1)(94 + 1)(92 + 1)(9 + 1)
964=(932+1).(932-1)
=(932+1)(916+1)(916-1)
=(932+1)(916+1)(98+1)(98-1)
=(932+1)(916+1)(98+1)(94+1)(94-1)
=(932+1)(916+1)(98+1)(94+1)(92+1)(92-1)
=(932+1)(916+1)(98+1)(94+1)(92+1)(9+1)(9-1)
Vì (932+1)(916+1)(98+1)(94+1)(92+1)(9+1)(9-1)>(932+1)(916+1)(98+1)(94+1)(92+1)(9+1)
=>964-1>(932+1)(916+1)(98+1)(94+1)(92+1)(9+1)
\(B=10^2+8^2+...+2^2-\left(9^2+7^2+5^2+3^2+1^2\right)\)
\(B=\left(10^2-9^2\right)+\left(8^2-7^2\right)+...+\left(2^2-1^2\right)\)
\(B=\left(10+9\right)\left(10-9\right)+\left(8+7\right)\left(8-7\right)+...+\left(2-1\right)\left(2+1\right)\)
\(B=19+15+...+3\)
Đến đây dễ rồi. Câu a) đang suy nghĩ
\(A=1+\left(5+1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+4\cdot\left(5+1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5-1\right)\left(5+1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5^{16}-1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)
\(4A=4+\left(5^{32}-1\right)\left(5^{32}+1\right)\)
\(4A=4+5^{64}-1\)
\(4A=5^{64}+3\)
\(A=\frac{5^{64}+3}{4}\)
\(=\frac{21.273.1333.4161.10101}{91.651.2451.6643.14763}\)
\(=\frac{3.7.13.21.31.43.73.57.91.111}{7.13.21.31.43.57.73.91.111.133}=\frac{3}{133}\)
Tuy nhiên cách làm trên phải có máy tính mới làm đc:
Có thể sử dụng công thức:
\(x^4+x^2+1=\left(x^2+x+1\right)\left(x^2-x+1\right)\)
Sau đó phân h:
\(2^4+2^2+1=\left(2^2+2+1\right)\left(2^2-2+1\right)=7.3\)
\(4^4+4^2+1=\left(4^2+4+1\right)\left(4^2-4+1\right)=21.13\)
....Tiếp tực làm thì sẽ ra đc kết quả:
\(=\frac{3.7.13.21.31.43.73.57.91.111}{7.13.21.31.43.57.73.91.111.133}=\frac{3}{133}\)
chịu