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?

1 Answer

abc=1872\sqrt2

Explanation:

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