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)   (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

     =  +  +  + .


It gives

        =  =  = .


So, the blanks are 81 and 33.     ANSWER

Solved.



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

Answer: 81 + 33*sqrt(6)

Work Shown
Using the binomial theorem

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











--------------------------------------------------------------------------

Here is another route which is a bit slower, but could be useful to know.
The FOIL rule can be used to confirm this or use the binomial theorem.

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






Then,




Let x = 3+sqrt(6)



Plug in x = 3+sqrt(6)






Therefore, 81 and 33 go in those blanks in the order mentioned.
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