Using only p&p, determine the simplest fractional form of this expression:
3333373 - 3333263
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3333373 + 6666633
The difference or sum of two cubes can be factored.
Numerator: (333337 - 333326)(333337^2 + (333337*333326) + 333326^2)
Denominator: (333337 + 666663)(333337^2 - (333337*666663) + 666663^2)
Numerator: 11 * (333337^2 + (333337*333326) + 333326^2)
Denominator: 1,000,000 * (333337^2 - (333337*666663) + 666663^2)
Claim - both of these expressions are equal:
(333337^2 + (333337*333326) + 333326^2)
=(333337^2 - (333337*666663) + 666663^2)
Call these expressions a^2 + ab + b^2 = a^2 - ac + c^2
ab + b^2 = - ac + c^2
ab + ac + b^2 - c^2 = 0
a(b+c) + (b-c)(b+c) = 0
(a+b-c)(b+c) = 0
In general, if a,b,c are all positive then b+c cannot be zero.
In our case, a+b-c = 333337 + 333326 - 666663 does equal zero.
So the answer is 11/1,000,000
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Posted by Larry
on 2023-09-04 10:33:02 |
Analytic solution
