An Interesting Algebra Puzzle | Simplify Without Calculators! 

Upon dividing the numerator and the denominator by 8, E= (2^98 +1)/(2^49 + 2^25 + 1) = 2^49 - 2^25 + 1.= 2(2^48-2^24) + 1 = 2(2^24)(2^24-1) + 1= 2(2^24)(2^12+1) (2^6+1)(2^3+1)7 + 1= 126(65)(4097)(2^24) + 1.

2^49-2^25+1= (128)^7-(32)^5+1

^ =read as to the power *=read as square root As per question (2^101 + 8)/(16^13 + 4^14 +8) Let's explain numerator (N) N=2^101 + 8 =8(2^98 +1)..{2^101=2^3×2^98)=8×2^98 Explain 2^98 +1 ={(2^49)^2}+(1)^2 ={(2^49)+1)}^2-{2×2^49×1} ={(2^49)+1)}^2-(2^50) ={(2^49+1)^2 -{(2^25)^2 ={2^49+2^25+1}{2^49 -2^25+1) Let {2^49+2^25+1}=a {2^49 -2^25+1}=b So, N=8ab Now explain denomiator (D) D=(16^13)+(4^14)+8 ={(2^4)^13}+{(2^2)^14}+8 =(2^52)+(2^28)+8 =8{2^49+2^25+1} =8a So, N/D =8ab/8a =b =2^49-2^25+1 (1/2)×2{2^49-2^25+1)} =(1/2){2^50-2.2^25+2} =(1/2)[{(2^25)^2 -2.1.2^25+(1^2)}+1] =(1/2){(2^25-1)^2 +1} =(1/2){(4064032-1)^2 +1} =(1/2){(4064031)^2 +1} =(1/2){16516347968961+1} =(1/2){16516347968962} =8258173984481 (may be)

2^10+2^3/2^3^3+2^2^2^2+2^3 2^2^5+1^1^1/1^1^3+1^1^1^1+1^1^1 1^2^1/1^3 2/3 (x ➖ 3x+2) .

562,949,919,866,881

(2^101+8)/(16^13+4^14+8)=562949919866881 It’s in my head.