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852+832+812+792+772-(842+822+802+782+762)
=852+832+812+792+772-842-822-802-782-762
=(852-842)+(832-822)+(812-802)+(792-782)+(772-762)
=(85-84)(85+84)+(83-82)(83+82)+(81-80)(81+80)+(79-78)(79+78)+(77-76)(77+76)
=85+84+83+...+77+76
=\(\frac{\left(85+76\right).\left(85-76+1\right)}{2}=805\)
\(a,=\left(93-7\right)\left(93+7\right)=100\cdot86=8600\\ b,=\left(45+55\right)^2=100^2=10000\)
a) \(A=123\left(123+154\right)+77^2\)
\(A=123^2+\left(123.154\right)+77^2=\left(123+77\right)^2=200^2=400\)
b) \(B=3^{24}-\left(2^{47}+1\right)\left(9^6-1\right)\)
\(B=3^{24}-\left(3^{12}-1\right)\left(3^{12}+1\right)\)
\(B=3^{24}-3^{24}+1=1\)
c) \(C=85^2+75^2+65^2+55^2-45^2-35^2-25^2-15^2\)
\(C=\left(85^2-15^2\right)+\left(75^2-25^2\right)+\left(65^2-35^2\right)+\left(55^2-45^2\right)\)
\(C=\left(85+15\right)\left(85-15\right)+\left(75+25\right)\left(75-25\right)+\left(65+35\right)\left(65-35\right)\left(55+45\right)\left(55-45\right)\)
\(C=100\left(60+50+40+30+20+10\right)\)
\(C=100.210=21000\)
\(a,=\left(50-1\right)\left(50+1\right)\)
\(=50^2-1\)
\(=2499\)
\(b,55^2-45^2\)
\(=\left(55-45\right)\left(55+45\right)\)
\(=10.100\)
\(=1000\)
\(\text{a) }49.51=\left(50-1\right)\left(50+1\right)=50^2-1^2=2500-1=2499\)
\(\text{b) }55^2-45^2=\left(55+45\right)\left(55-45\right)=100.10=1000\)
a) 51x49
=( 50+1) x(50-1)
=50x50 -1x1
=2500-1=2499
b) 822-812
=(82-81)x (82+81)
=1x 163
=163
B=\(3^{24}-\left(27^4+1\right)\left(9^6+1\right)=-1062883\)A=123(123+154)+772=123.277+772=34843
C=\(85^2+75^2+65^2+55^2-45^2-35^2-25^2-15^2=16000\)
\(=85^2-84^2+83^2-82^2+81^2-80^2+79^2-78^2+77^2-76^2\)
=85+84+...+77+76
=10*(85+76)/2=805
a> 529+5929+3542=10000
B. 6084+784-4368=2500
c>2451
d>8952064
\(=\left(80-75\right)\left(80+75\right)+\left(70-65\right)\left(75+65\right)+...+\left(10-5\right)\left(10+5\right)\)
\(=5\cdot155+5\cdot135+..+5\cdot15\)
\(=5\cdot\left(155+135+...+15\right)\)
\(=5\cdot\left(155+15+135+35+115+55+95+75\right)\)
\(=5\cdot\left(170\cdot4\right)\)
\(=5\cdot680=3400\)
a: \(78\cdot82=80^2-4=6400-4=6396\)
b: \(45^2+90\cdot55+55^2=100^2=10000\)