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(83−6)(43+6)(8\sqrt{3}-\sqrt{6})(4\sqrt{3}+\sqrt{6})

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Full solution

Q. (83−6)(43+6)(8\sqrt{3}-\sqrt{6})(4\sqrt{3}+\sqrt{6})
  1. Identify Expression: Identify the expression after distributing (43+6)(4\sqrt{3} + \sqrt{6}). Distribute (43+6)(4\sqrt{3} + \sqrt{6}) to each term of (83−6)(8\sqrt{3} - \sqrt{6}). (83−6)(43+6)=83(43+6)−6(43+6)(8\sqrt{3} - \sqrt{6})(4\sqrt{3} + \sqrt{6}) = 8\sqrt{3}(4\sqrt{3} + \sqrt{6}) - \sqrt{6}(4\sqrt{3} + \sqrt{6})
  2. Distribute Terms: Simplify 83(43+6)8\sqrt{3}(4\sqrt{3} + \sqrt{6}).
    83(43+6)8\sqrt{3}(4\sqrt{3} + \sqrt{6})
    =83(43)+83(6)= 8\sqrt{3}(4\sqrt{3}) + 8\sqrt{3}(\sqrt{6})
    =32×3+83×6= 32 \times 3 + 8\sqrt{3 \times 6}
    =96+818= 96 + 8\sqrt{18}
    =96+8×32= 96 + 8 \times 3\sqrt{2}
    =96+242= 96 + 24\sqrt{2}
  3. Simplify First Term: Simplify −6(43+6)-\sqrt{6}(4\sqrt{3} + \sqrt{6}).
    −6(43+6)-\sqrt{6}(4\sqrt{3} + \sqrt{6})
    = −6(43)−6(6)-\sqrt{6}(4\sqrt{3}) - \sqrt{6}(\sqrt{6})
    = −418−6-4\sqrt{18} - 6
    = −4×32−6-4 \times 3\sqrt{2} - 6
    = −122−6-12\sqrt{2} - 6
  4. Simplify Second Term: Combine the results from Step 22 and Step 33.\newline96+242−122−696 + 24\sqrt{2} - 12\sqrt{2} - 6\newline= 96−6+242−12296 - 6 + 24\sqrt{2} - 12\sqrt{2}\newline= 90+(24−12)290 + (24 - 12)\sqrt{2}\newline= 90+12290 + 12\sqrt{2}

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