Chứng minh các đẳng thức sau:
a) (2 + 1) (22 + 1) (24 + 1) (28 + 1) (216 + 1) = 232 - 1
b) 1002 + 1032 + 1052 + 942 = 1012 + 982 + 962 + 1072
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14 tháng 7 2021
a) Ta có: \(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)
\(=2^{32}-1\)
PN
1
19 tháng 6 2021
`A=(2-1)(2+1)(2^2+1)...(2^16+1)`
`=(2^2-1)(2^2+1)....(2^16+1)`
`=(2^4-1)....(2^16+1)`
`=2^32-1<2^32`
`=>A<B`
MT
9 tháng 7 2015
a,(a-1)(a-2)+(a-3)(a-4)-(2a^2+5a+34)
=a2-3a+2+a2-7a+12-2a2-5a-34
=-15a-20
sai đề kakakakakaka
b, (a-b)(a^2+ab+b^2)-(a+b)(a^2+ab+b^2)
=a3-b3-(a3+b3)
=a3-b3-a3-b3
=-2b3
13 tháng 5 2022
a: A=(100-99)(100+99)+(98-97)(98+97)+...+(2-1)(2+1)
=100+99+98+...+2+1
=5050
b: \(B=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\cdot...\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^4-1\right)\left(2^4+1\right)\cdot...\cdot\left(2^{64}+1\right)+1\)
\(=\left(2^8-1\right)\left(2^8+1\right)\cdot...\cdot\left(2^{64}+1\right)\)+1
\(=2^{64}-1+1=2^{64}\)
13 tháng 10 2021
\(2,\\ a,a^3+b^3=a^3=3a^2b+3ab^2+b^3-3a^2b-3ab^2\\ =\left(a+b\right)^3-3ab\left(a+b\right)\\ b,a^3+b^3+c^3-3abc\\ =\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc\\ =\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)\\ =\left(a+b+c\right)\left(a^2+b^2+c^2-ac-ab-bc\right)\)
\(\left(2-1\right)\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^2-1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^4-1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^8-1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)
\(=\left(2^{16}-1\right)\left(2^{16}+1\right)\)
\(=2^{32}-1\)
b/ \(100^2+\left(100+3\right)^2+\left(100+5\right)^2+\left(100-6\right)^2\)
\(=100^2+100^2+100^2+100^2+4.100+9+25+36\)
\(=100^2+2.100+1+100^2-4.100+4+100^2-8.100+16+100^2+14.100+49\)
\(=\left(100+1\right)^2+\left(100-2\right)^2+\left(100-4\right)^2+\left(100+7\right)^2\)
Thanks