Radical Expression

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5-3 3 \sqrt{3} 5+3 3 \sqrt{3} 5+4 3 \sqrt{3} 6-5 3 \sqrt{3}

Erik Wang by Erik Wang , 20, China

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1 solution

( 2 + 3 + 6 ) ( 2 − 3 + 6 ) = ( ( 2 + 6 ) + 3 ) ( ( 2 + 6 ) − 3 ) (\sqrt2+\sqrt3+\sqrt6)(\sqrt2-\sqrt3+\sqrt6)\\=((\sqrt2+\sqrt6)+\sqrt3)((\sqrt2+\sqrt6)-\sqrt3) Apply the difference of two squares identity: ( a + b ) ( a − b ) = a 2 − b 2 (a+b)(a-b)=a^2-b^2 : ( 2 + 6 ) 2 − ( 3 ) 2 = 2 + 6 + 2 2 × 6 − 3 = 8 + 2 12 − 3 = 5 + 2 4 × 3 = 5 + 4 3 (\sqrt2+\sqrt6)^2-(\sqrt3)^2=2+6+2\sqrt{2\times6}-3\\=8+2\sqrt{12}-3\\=5+2\sqrt4\times\sqrt3\\\boxed{=5+4\sqrt{3}}

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