Question

Aratati ca urmatoarele numere sunt patrate unor nr. naturale a)5 la puterea 40 b 23 la puterea 100 c) (7 la puterea 6)la puterea 5 d) 2 la puterea 18 ori 3 la puterea 6 e)81 la puterea 7 ori 7 la puterea 4 f)200+2 ori(1+2+......+199) g) 3 la puterea 23 ori 4 la puterea 23 - 2 la puterea12 ori 6 la puterea 23 h) 3 la puterea 2011+2 ori 3 la puterea 2010+3 la puterea 2009+3 la puterea 2008

Answer 1

a)5^40
b) 23^100 =(23^50)^2
c) (7^6)^5 =7^30 =(7^15)^2
d) 2^18 ×3^6 =(2^9 ×3^3)^2
e)81^7 × 7^4 =(9^2)^7 × 7^4 =(9^7 × 7^2)^2
f)200+2 ×(1+2+......+199) =200+2 ×199×200:2=200+199×200=
=200(1+199)=200×200=200^2
g) 3^23 × 4^23 - 2^21 × 6^23 =3^23 × 2^46 - 2^21 × 2^23× 3^23 =
=3^23 × 2^46 - 2^46× 3^23 =0
h) 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