Alegrbra in 8th grade help!!! A^3 + B^3 = 1729 and C^3 + D^3 = 1729. What is the sum of A + B + C + D?

Question

please help me I know the answers but don't know how to find it Help!!!!!

Answer 1

A^3 + B^3 = 1729 and C^3 + D^3 = 1729. What is the sum of A + B + C + D?




a^3 +b^3 = 1728 +1 = 12^3 +1^3
a =12 ; b =1
a+b =12 +1 =13

c^3 +d^3 = 12^3 +1^3
c+d = 12 +1 =13

a+b +c +d =26

Answer 2

You may be missing some information. You need an equation for each variable that needs defined. (i.e. to determine the value of A and B in the first equation you would need 2 equations.) You have 4 variables and only 2 equations.
Given the limited information you have, there are a great number of solutions. (Starting with A=0, B=1729^(1/3). A=1, B=(1728)^(1/3)....A=(1729)^(1/3), B=0)
I would see if there aren't additional restrictions on the variables or perhaps other equations like A=C or A+B=(insert value).

Answer 3

I would cube root A^3 + B^3 = 1729 and C^3 + D^3 = 1729.
So A^3 + B^3 = 1729 and C^3 + D^3 = 1729. So ³√A^3+³√B^3=³√1729 and ³√C^3+³√D^3=³√1729. This gives us A+B=³√1729 and C+D=³√1729. Therefore A+B+C+D=2*(³√1729)=24.0046(6dp)

Answer 4

1729 can be written as 12³ + 1³ or 10³ + 9³.
12³+1³ = (12+1)(144-12+1) = 13*133
and
10³+9³ = (10+9)(91) = 19*91
So we can take A+B = 13 and C+D = 19
and A+B+C+D = 32.
BTW 1729 is the smallest positive integer
that can be written as the sum of 2 cubes in 2
different ways(As Ramanujan once pointed out to Hardy).