Question

scrieti ca o singura putere: a) 2^1×2^2×2^3×.........×2^100;b) 5^1×5^3×5^5×.....×5^19;c) 7^1×7^4×7^7×.....7^301.

Answer 1

a)2^1×2^2×2^3×.........×2^100 =2⁵⁰⁵⁰
1+2+3+...+100=100×101/2=5050

b) 5^1×5^3×5^5×.....×5^19=5¹⁰⁰
1+3+5+...+19=10²
2n-1=19   n=10 

c) 7^1×7^4×7^7×.....7^301 =7¹⁵²⁵¹
1+4+7+...+301=101(3×101-1)/2=101(303-1)/2=101×302/2=101×151=
1+4+7+...+3n-2=n(3n-1)/2
3n-2=301    n=101

Answer 2

a)2^(1+2+3+...+100)=2^(100×101÷2)=2^5050
b)5^(1+3+5+...+19)=5^(2×0+1+2×1+1+2×2+1...+2×9+1)=5^[2×(1+2+3+...+9)+1×10]=5^(2×9×10÷2+10)=5^(90+10)=5^100
c)7^(1+4+7+...+301)=7^(3×0+1+3×1+1+3×2+1+...+3×100+1)=7^[2×(1+2+3+...+100)+1×101]=7^(2×100×101÷2+101)=7^(10100+101)=7^10201