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

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.

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