Question

Aratati ca nr n este patrat perfect a) n=27^9×32^11:2-16^6×2×6^27 b) n=2^2011-2^2010-2^2009-2^2008 c) n=3^23×4^23-2^21×6^23 b) n=3^2011+2×3^2010+3^2009+3^2008
Dau coroana :*

Answer 1

a) n=27^9×32^11:2-16^6×2×6^27 =
=(3^3)^9×(2^5)^11:2-(2^4)^6×2×2^27×3^27 =
=3^27×2^55:2-2^24×2^28×3^27 =
=3^27×2^54-2^52×3^27 =
=3^27×2^52×(2^2-1)=
=3^27×2^52×(4-1)=
=3^27×2^52×3=
=3^28×2^52=
=(3^14×2^26)^2

b) n=2^2011-2^2010-2^2009-2^2008 =
=2^2008×(2^3-2^2-2^1-1) =
=2^2008×(8-4-2-1) =
=2^2008×1 =
=(2^1004)^2


c) n=3^23×4^23-2^21×6^23 =
=3^23×2^46-2^21×2^23×3^23 =
=3^23×2^46-2^44×3^23 =
=3^23×2^44×(2^2-1) =
=3^23×2^44×(4-1) =
=3^23×2^44×3 =
=3^24×2^44 =
=(3^12×2^22)^2

d) n=3^2011+2×3^2010+3^2009+3^2008=
=3^2008×(3^3+2×3^2+3^1+3^0)=
=3^2008×(27+18+3+1)=
=3^2008×49=
=3^2008×7^2=
=(3^1004×7)^2=