Question

Evaluate\( ( √3 +√2) ^6 - ( √3 -√2) ^6.\)

Answer 1

Given:

(√3 + √2)⁶ − (√3 − √2)⁶

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Step 1: Let

x = √3 + √2,    y = √3 − √2

Required value = x⁶ − y⁶

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Step 2: Find x + y and xy

x + y = 2√3

xy = (√3 + √2)(√3 − √2) = 3 − 2 = 1

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Step 3: Use identity

x⁶ − y⁶ = (x − y)(x + y)(x² + y²)(x² − xy + y²)

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Step 4: Compute required terms

x − y = 2√2

x² + y² = (x + y)² − 2xy = (2√3)² − 2 = 12 − 2 = 10

x² − xy + y² = (x² + y²) − xy = 10 − 1 = 9

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Step 5: Substitute values

x⁶ − y⁶ = (2√2)(2√3)(10)(9)

= 4√6 × 90

= 360√6

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Final Answer:

(√3 + √2)⁶ − (√3 − √2)⁶ = 360√6