Question

Question 11 Find (a + b)^4 – (a – b)^4. Hence, evaluate [ 3^(1/2) + 2^(1/2) ]^4 - [ 3^(1/2) - 2^(1/2) ]^4

Class X1 - Maths -Binomial Theorem Page 167

Answer 1

(a + b)⁴
use the formula,
(x + y)^n = nC0.x^n + nC1.x^(n-1)y + nC2.x^(n-2)y^2 + ........+nCn.y^n

(a + b)^4 = 4C0.a^4 + 4C1.a^3.b + 4C2a^2.b^2 + 4C3.a.b^3 + 4C4.b^4
= {4!/4!}a^4 + {4!/3!}a^3b + {4!/2!×2!}a^2b^2 + {4!/3!}ab^3 + {4!/4!}b^4
= a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

similarly (a - b)⁴
use the formula,
(x - y)^n = nC0.x^n - nC1.x^(n-1)y + nC2.x^(n-2)y^2 - ........(-1)^n.nCn.y^n

(a - b)^4 = 4C0a^4 - 4C1.a^3b + 4C2.a^2b^2 - 4C3.ab^3 + 4C4.b^4
= a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴

now,
(a + b)⁴ - (a -b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab² + b⁴ -(a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴)
=8a³b + 8ab³
= 8ab(a² + b²)

hence,
[√3 + √2 ]⁴ - [ √3 - √2 ]⁴ = 8 × √3 × √2 [√3² + √2²]
= 8 × √6 [ 3 + 2]
= 8 × √6[5]
= 40√6

Answer 2

Answer:

hope this helps u.....