Find (a + b)⁴ - (a - b)⁴. Hence, evaluate (√3 + √2)⁴ - (√3 - √2)⁴

Solution:

Using binomial theorem, we will evaluate the expressions (a + b)⁴ and (a - b)⁴.

• (a + b)⁴ = ⁴C₀ a⁴ + ⁴C₁ a³b + ⁴C₂ a²b² + ⁴C₃ ab³ + ⁴C₄ b⁴
   
• (a - b)⁴ = ⁴C₀ a⁴ - ⁴C₁ a³b + ⁴C₂ a²b² - ⁴C₃ ab³ + ⁴C₄ b⁴

Therefore,

(a + b)⁴ - (a - b)⁴ = [(⁴C₀ a⁴ + ⁴C₁ a³b + ⁴C₂ a²b² + ⁴C₃ ab³ + ⁴C₄ b⁴) - (⁴C₀ a⁴ - ⁴C₁ a³b + ⁴C₂ a²b² - ⁴C₃ ab³ + ⁴C₄ b⁴)]

= 2( ⁴C₁ a³b + ⁴C₃ ab³)

= 2(4a³b + 4ab³)

= 8ab (a² + b²)

Putting a = √3 and b = √2

(√3 + √2)⁴ - (√3 - √2)⁴ = 8(√3)(√2) ((√3)² + (√2)²)

= 8√6 [3 + 2]

= 40√6

NCERT Solutions Class 11 Maths Chapter 8 Exercise 8.1 Question 11

Find (a + b)⁴ - (a - b)⁴. Hence, evaluate (√3 + √2)⁴ - (√3 - √2)⁴

Summary:

Using the binomial theorem, (a + b)⁴ - (a - b)⁴ = 8ab (a² + b²). Hence, (√3 + √2)⁴ - (√3 - √2)⁴ = 40√6