Algebra/Surds

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Question
Express 4/(2+√3+√7) in the form 1 + a√x + b√y where x and y are integers.

Answer
Rationalization is required in this instance, and  (a+b)(a-b)= a^2 -b^2 shall be used.

4/(2+√3+√7)= 4/(2+√3+√7) * [(2+√3)-√7]/[(2+√3)-√7]     

         = 4[(2+√3)-√7]/[ (2+√3)^2 - (√7)^2]        

         = 4[(2+√3)-√7]/[ 4 + 4√3 + 3 - 7]

         = 4[(2+√3)-√7]/4√3

         = [(2+√3)-√7]/√3

         = [(2+√3)-√7]/√3  *  √3/√3

         =  √3[(2+√3)-√7]/ 3

         =  2√3/3 +  1  - √21/3

         =  √12/3 + 1 - √21/3

         =  1 - 1/3*√21 + 1/3*√12  (shown)

Hope this helps. Peace.  

Algebra

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Frederick Koh

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I can answer questions concerning calculus, complex numbers, vectors, statistics , algebra and trigonometry for the O level, A level and 1st/2nd year college math/engineering student.

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