An Amazing Trick to Simplify Without Calculator!

[@backgammonmaster]

Dear sir, If you take a=2 ^25 THEN THE NUMERATOR CAN BE SIMPLIFIED BY USING Sophie Gemain identity and simplify. it can be done under a minute, no need to take 8 and a half minute and make it so complicated.Thank you.

[@davidseed2939]

quick solution let a=2^25
E=(4a^4+1)/(2a^2+2a+1)
Now the trick here is to follow the line that you use for rationalising the denominator by multiplying by the conjugate. In this case we multiply top and bottom by (2a^2-2a+1)
Now (2a^2+2a+1)*(2a^2-2a+1) = 4a^4+1
So E = 2a^2-2a+1
2^51-2^26+1
A ridiculous question since with numbers that large you still dont have a numerical value
if instead a=512 then it would be reduced to 2a(a-1) +1= 1024(511) +1
which you could evaluate in an exam

[@Quest3669]

2^51-2^26+1

[@RealQinnMalloryu4]

1^101+/1^51+1^16 10^10^1+/2^3+3^3 /2^1+1^3 /2+3 (x ➖ 3x+2)

[@SidneiMV]

E = (2¹⁰² + 1)/(2⁵¹ + 2²⁶ + 1)
E = [(2⁵¹ + 1)² - 2⁵²)]/(2⁵¹ + 1 + 2²⁶)
*E = (2⁵¹ + 1 - 2²⁶)*
or E = 2⁵⁰ + (2²⁵ - 1)²

[@asvquickcalculations6712]

2^102+1=2^102+2*2^51 8:37 +1-2*2^51=(2^51+1)^2-(2^26)^2=(2^51+2^26+1)(2^51+1-2^26)
Why are you making simple thing as complicated .

[@adribber]

Let, (2¹⁰² + 1)/(2⁵¹ + 2²⁶ + 1) = k
k = (2²*2¹⁰⁰ + 1)/(2*2⁵⁰ + 2*2²⁵ + 1)
= (2²*2¹⁰⁰ + 1)/{2(2⁵⁰ + 2²⁵) + 1}
Let, x = 2²⁵
k = (4x⁴ + 1)/{2(x² + x) + 1}
= {(2x² + 1)² - 4x²}/(2x² + 2x + 1)
= {(2x² + 1) + 2x}{(2x² + 1) - 2x}/(2x² + 2x + 1)
= (2x² + 2x + 1)(2x² - 2x + 1)/(2x² + 2x + 1)
= 2x² - 2x + 1
Recall, x = 2²⁵
k = 2x² - 2x + 1
= 2(2²⁵)² - 2(2²⁵) + 1
= 2(2⁵⁰) - 2²⁶ + 1
= 2⁵¹ - 2²⁶ + 1

[@ManojkantSamal]

(2^51)-(2^26)+1.....Much more simplifications are there.
Explain later

[@RashmiRay-c1y]

The given expression E=2.718 (4a^4+1)/(2a^2+2a+1) = 2a^2-2a+1, where a = 2^25. So, E =2.718 2^51-(2^26 -1) =2.252×10¹⁵ 2^51 - (2^13+1)(2^13-1) =2.252×10¹⁵ 2^51 - (8191)(8193) = 2251799746576385.