evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

 ncert cbse chapter 8 binomial theorem miscellaneous exercise 

5.evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

 (a+b)^6 = {a^6} + C(6,1){a^5}{b} +C(6,2){a^4}{b^2}+C(6,3){a^3}{b^3} +C(6,4){a^2}{b^4}++C(6,5){a^1}{b^5}+C(6,6)(b)^6

 C(6,1) = 6 =C(6,5)

C(6,2)=15=C(6,4)

C(6,3)=20

C(6,6)=1

(a+b)^6 = {a^6} + 6{a^5}{b} +15{a^4}{b^2}+20{a^3}{b^3} +15{a^2}{b^4}+6{a^1}{b^5}+(b)^6

 (a-b)^6 = {a^6} - 6{a^5}{b} +15{a^4}{b^2} - 20{a^3}{b^3} +15{a^2}{b^4}- 6{a^1}{b^5}+(b)^6

adding the two equations

{ (a+b)^6} - {(a-b)^6} =

2[6{a^5}{b} +20{a^3}{b^3}+ 6{a^1}{b^5} ]

a=sqrt(3)

b=sqrt(2)

{ (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

= 2[6*{(sqrt(3))^5}{(sqrt(2))} +20{(sqrt(3))^3}{(sqrt(2))^3}+6{(sqrt(3))^1}{(sqrt(2))^5} ]

=2[54+120+24 ] {sqrt(6)}

=396*sqrt(6)

chapter 8 binomial theorem miscellaneous exercise

 1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375

solution

 

2.  Find a if the coefficients of (x^2)  & (x^3) in the expansion of {(3+ax)^9} are equal 

solution

  3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}

solution

 

5.evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }

solution 

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