SOLUTION: Fill in the blanks with positive integers: (3 + sqrt(6))^3 = ___ + ___ * sqrt(6)

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Question 1208689: Fill in the blanks with positive integers:
(3 + sqrt(6))^3 = ___ + ___ * sqrt(6)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Fill in the blanks with positive integers:
(3 + sqrt(6))^3 = ___ + ___ * sqrt(6)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Use the identity

    %28a%2Bb%29%5E3 = a%5E3 + 3a%5E2%2Ab + 3a%2Ab%5E2 + b%5E3.


It gives

       %283+%2B+sqrt%286%29%29%5E3 =  = 27+%2B+27%2Asqrt%286%29+%2B+9%2A6+%2B+6%2Asqrt%286%29 = 81+%2B+33%2Asqrt%286%29.


So, the blanks are 81 and 33.     ANSWER

Solved.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 81 + 33*sqrt(6)

Work Shown
%28a%2Bb%29%5E3+=+a%5E3+%2B+3a%5E2b%2B3ab%5E2+%2B+b%5E3 Using the binomial theorem

Plug in a = 3 and b = sqrt(6)



%283%2Bsqrt%286%29%29%5E3+=+27+%2B+27%2Asqrt%286%29+%2B+54+%2B+6%2Asqrt%286%29

%283%2Bsqrt%286%29%29%5E3+=+%2827%2B54%29+%2B+%2827%2Asqrt%286%29%2B6%2Asqrt%286%29%29

%283%2Bsqrt%286%29%29%5E3+=+81+%2B+%2827%2B6%29%2Asqrt%286%29

%283%2Bsqrt%286%29%29%5E3+=+red%2881%29+%2B+red%2833%29%2Asqrt%286%29

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Here is another route which is a bit slower, but could be useful to know.
%28a%2Bb%29%5E2+=+a%5E2%2B2ab%2Bb%5E2 The FOIL rule can be used to confirm this or use the binomial theorem.

%283%2Bsqrt%286%29%29%5E2+=+3%5E2+%2B+2%2A3%2Asqrt%286%29+%2B+%28sqrt%286%29%29%5E2 Plug in a = 3 and b = sqrt(6)

%283%2Bsqrt%286%29%29%5E2+=+9+%2B+6%2Asqrt%286%29+%2B+6

%283%2Bsqrt%286%29%29%5E2+=+15+%2B+6%2Asqrt%286%29


Then,
%283%2Bsqrt%286%29%29%5E3+=+%283%2Bsqrt%286%29%29%283%2Bsqrt%286%29%29%5E2

%283%2Bsqrt%286%29%29%5E3+=+%283%2Bsqrt%286%29%29%2815+%2B+6%2Asqrt%286%29%29

%283%2Bsqrt%286%29%29%5E3+=+x%2815+%2B+6%2Asqrt%286%29%29 Let x = 3+sqrt(6)

%283%2Bsqrt%286%29%29%5E3+=+x%2A15+%2B+x%2A6%2Asqrt%286%29

Plug in x = 3+sqrt(6)



%283%2Bsqrt%286%29%29%5E3+=+45+%2B+15%2Asqrt%286%29%2B18%2Asqrt%286%29+%2B+36

%283%2Bsqrt%286%29%29%5E3+=+red%2881%29+%2B+red%2833%29%2Asqrt%286%29
Therefore, 81 and 33 go in those blanks in the order mentioned.