Question

How do you use the binomial series to expand `(3a+4b)^4`?

Answer 1

`81a^4 +432ba^3 +864b^2a^2 +768b^3 a+256b^4`

Explanation:

Binomial expansion of `(x+a)^n=x^n +nC1 ax^(n-1) +nC2 a^2x^(n-2) +.....`

Now use n=4, x= 3a , a =4b and write the expansion

`(3a+4b)^4= (3a)^4 +4C1 (4b) (3a)^3 + 4C2 (4b)^2 (3a)^2 +4C3 (4b)^3 (3a) + 4C4 (4b)^4`

=`81a^4 +432ba^3 +864b^2a^2 +768b^3 a+256b^4`

[4C1=4; 4C2=`(4.3)/(1.2)=6`; 4C3=#(4.3.2)/(1.2.3)=4;

4C4=`(4.3.2.1)/(1.2.3.4)=1`]