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4 (1+ 8 ) (1+ 3 8 ) (1+ 5 8 ) (1+ 7 8 ) is equal to

Options

  1. A1
  2. B1 8
  3. C1 4
  4. D1 2

Correct answer

D. 1 2

Step-by-step solution

Let the given expression be E = 4 (1 + 8 ) (1 + 3 8 ) (1 + 5 8 ) (1 + 7 8 ) . Using the identity ( - ) = - , we have 7 8 = - 8 and 5 8 = - 3 8 . Substituting these into the expression: E = 4 (1 + 8 ) (1 + 3 8 ) (1 - 3 8 ) (1 - 8 ) Grouping the terms: E = 4 (1 - ^2 8 ) (1 - ^2 3 8 ) E = 4 ^2 8 ^2 3 8 Using the identity ^2 = 1 - 2 2 : E = 4 ( 1 - ( /4) 2 ) ( 1 - (3 /4) 2 ) E = 4 ( 1 - 1/ 2 2 ) ( 1 - (-1/ 2 ) 2 ) E = 4 ( 1 - 1/ 2 2 ) ( 1 + 1/ 2 2 ) E = 4 ( 1 - 1/2 4 ) = 4 ( 1/2 4 ) = 4 1 8 = 1 2 . Answer: 1 2

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