Find \((a + b)^4 - (a - b)^4\). Hence, evaluate \((\sqrt3 + \sqrt2)^4 - (\sqrt3 - \sqrt2)^4\).

Question

Using Binomial Theorem, evaluate \((102)^5\).

Answer 1

Part I: Find (a + b)⁴ − (a − b)⁴

Step 1: Expand using Binomial Theorem

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴

Step 2: Subtract

(a + b)⁴ − (a − b)⁴

= 8a³b + 8ab³

= 8ab(a² + b²)

Result:

(a + b)⁴ − (a − b)⁴ = 8ab(a² + b²)

Part II: Evaluate (√3 + √2)⁴ − (√3 − √2)⁴

Step 1: Substitute a = √3, b = √2

a² + b² = 3 + 2 = 5

ab = √6

Step 2: Evaluate

8ab(a² + b²) = 8 × √6 × 5

= 40√6

Final Answers:

(a + b)⁴ − (a − b)⁴ = 8ab(a² + b²)

(√3 + √2)⁴ − (√3 − √2)⁴ = 40√6

Solution and Explanation