Question

How do you expand `(2x+3y)^4`?

Answer 1

`16x^4+96x^3y+216x^2y^2+216xy^3+81y^4`

Explanation:

`"using the "color(blue)"Binomial theorem"`

`•color(white)(x)(a+b)^n=sum_(r=0)^n((n),(r))a^(n-r)b^r`

`"where "((n),(r))=(n!)/(r!(n-r)!)`

`"we can also generate the binomial coefficients using"`
`"the appropriate row of "color(blue)"Pascal's triangle"`

`"for "n=4to1color(white)(x)4color(white)(x)6color(white)(x)4color(white)(x)1`

`"here "a=2x" and "b=3y`

`rArr(2x+3y)^4`

`=1.(2x)^4(3y)^0+4.(2x)^3(3y)^1+6.(2x)^2(3y)^2+4.(2x)^1(3y)^3+1.(2x)^0(3y)^4`

`=16x^4+96x^3y+216x^2y^2+216xy^3+81y^4`