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[@als2cents679]

you can simplify the calculations by a lot by using (a+b)(a-b) formula, by choosing x the middle of all those numbers, i.e. x = 501.5
answer = a = sqrt(500 * 501 * 502 * 503 + 1)
a^2 = (x - 3/2) (x - 1/2) (x + 1/2) (x + 3/2) + 1
= (x^2 - 9/4) (x^2 - 1/4) + 1
use y = x^2 to get
a^2 = y^2 - (10/4) y + 9/16 + 1
a^2 = (16 y^2 - 40 y + 25) / 16
a^2 = (4 y - 5)^2 / 4^2
a^2 = ((4 y - 5) / 4)^2
a = (4 y - 5) / 4
a = (4 x^2 - 5) / 4
a = ((2 x)^2 - 5) / 4
a = ((2 * 501.5)^2 - 5) / 4
a = (1003^2 - 5) / 4
a = (1000000 + 6000 + 9 - 5) / 4
a = 250000 + 1500 + 1
a = 251501

[@nasrullahhusnan2289]

Consider x=a(a+1)(a+2)(a+3)
=(a²+3a)(a²+3a+2)
=(k-1)(k+1)
=k²-1
where k=a²+3a+1 --> x+1=k²
sqrt(x+1)=k
=a²+3a+1
=a(a+3)+1
Letting a=500, sqrt(x+1) is the quantity we have to find.
sqrt(x+1)=500(503)+1
=251501

[@RyanLewis-Johnson-wq6xs]

Input
sqrt(500×501×502×503 + 1) = 251501
Result
True 251501

[@RyanLewis-Johnson-wq6xs]

I did it in my head.

[@erwinkurniadi1850]

4:09
Is there a difference between,
x²+3x+1 (x²+3x+1) and
(x²+3x+1)(x²+3x+1)?
(x²+3x+1)(x²+3x+1)=(x²+3x+1)²
x²+3x+1(x²+3x+1)=...?

[@YAWTon]

Yes, there is a difference. He forgot to write the parentheses...

[@riccardofroz]

sqrt(500(501)(502)(503)+1)
x=500
sqrt(x(x+1)(x+2)(x+3)+1)=
sqrt(x(x^2+3x+2)(x+3)+1)=
sqrt(x(x^3+6x^2+11x+6)+1)=
sqrt(x^4+6x^3+11x^2+6x+1)=
sqrt(x^4+4x^3+6x^2+4x+1+2x^3+5x^2+2x)=
sqrt((x+1)^4+2x^3+5x^2+2x)=
sqrt((x+1)^4+2x(x^2+2x+1)+x^2)=
sqrt((x+1)^4+2x(x+1)^2+x^2)=
sqrt(((x+1)^2+x)^2)=
(x+1)^2+x=
250000+1000+1+500=
251501