Question

evaluate (2+√3)^7 + (2 - √3)^7 by binomial theorem

Answer 1

let's solve your question

evaluate (2+√3)^7 + (2 - √3)^7

(2+√3)7+(2−√3)7

=10084

Answer:
10084

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Answer 2

(2+√3)⁷  +  (2-√3)⁷ = 10084

Step-by-step explanation:

(x + y)ⁿ  = ⁿC₀xⁿy⁰ + ⁿC₁xⁿ⁻¹y¹  +...........................+ ⁿCₙ₋₁x¹yⁿ⁻¹+ ⁿCₙx⁰yⁿ

(2+√3)⁷ = ⁷C₀2⁷(√3)⁰ + ⁷C₁2⁶(√3)¹ +  ⁷C₂2⁵(√3)² + ⁷C₃2⁴(√3)³  + ⁷C₄2³(√3)⁴ +  ⁷C₅2²(√3)⁵ + ⁷C₆2¹(√3)⁶ +  ⁷C₇2⁰(√3)⁷

(2-√3)⁷ = ⁷C₀2⁷(-√3)⁰ + ⁷C₁2⁶(-√3)¹ +  ⁷C₂2⁵(-√3)² + ⁷C₃2⁴(-√3)³  + ⁷C₄2³(-√3)⁴ +  ⁷C₅2²(-√3)⁵ + ⁷C₆2¹(-√3)⁶ +  ⁷C₇2⁰(-√3)⁷

=> (2-√3)⁷ = ⁷C₀2⁷(√3)⁰ - ⁷C₁2⁶(√3)¹ +  ⁷C₂2⁵(√3)² - ⁷C₃2⁴(√3)³  + ⁷C₄2³(√3)⁴ -  ⁷C₅2²(√3)⁵ + ⁷C₆2¹(√3)⁶ -  ⁷C₇2⁰(√3)⁷

Adding Both

(2+√3)⁷  +  (2-√3)⁷ = 2(⁷C₀2⁷(√3)⁰  +  ⁷C₂2⁵(√3)²  + ⁷C₄2³(√3)⁴ + ⁷C₆2¹(√3)⁶ )

=> (2+√3)⁷  +  (2-√3)⁷ = 2(128  +  21*32 * 3  + 35*8* 9 + 7*2*27 )

=> (2+√3)⁷  +  (2-√3)⁷ = 2(128  +  2016  + 2520 + 378 )

=> (2+√3)⁷  +  (2-√3)⁷ = 2(5042 )

=>  (2+√3)⁷  +  (2-√3)⁷ = 10084

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