The number #sqrt(104sqrt6+468sqrt10+144sqrt15+2006# can be written as #asqrt2+bsqrt3+csqrt5#, where a, b, and c are positive integers. Compute the product abc?

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Answer 1 · 2018-06-29T02:08:00

#abc=1872\sqrt2#

Given that

#\sqrt{104\sqrt6+468\sqrt10+144\sqrt15+2006}=a\sqrt2+b\sqrt3+c\sqrt5#

#104\sqrt6+468\sqrt10+144\sqrt15+2006=(a\sqrt2+b\sqrt3+c\sqrt5)^2#

#104\sqrt6+468\sqrt10+144\sqrt15+2006=2a^2+3b^2+5c^2+ab\sqrt6+ac\sqrt10+bc\sqrt15#

By comparing the coefficients of #\sqrt2, \sqrt3# & #\sqrt5# on both the sides we get

#ab=104#
#ac=468#
#bc=144#

Multiplying above three equations, we get

#ab\cdot ac\cdot bc=104\cdot 468\cdot 144#

#(abc)^2=104\cdot 468\cdot 144#

#abc=\sqrt{104\cdot 468\cdot 144}#

#abc=12\cdot156\sqrt2#

#abc=1872\sqrt2#