Question

How do you use the binomial theorem to expand and simplify the expression `(a+5)^5`?

Answer 1

See answer below

Explanation:

The binomial theorem states that the expansion of `(a+b)^n=((n),(0))a^nb^0+((n),(1))a^(n-1)b+...+((n),(n))a^0b^n, n in NN`

`(a+5)^5=((5),(0))a^5+((5),(1))a^4*5+((5),(2))a^3*5^2+((5),(3))a^2*5^3+((5),(4))a^1*5^4+((5),(5))a^0*5^5`
`=a^5+25a^4+250a^3+1250a^2+3125a+3125`

Answer 2

`(a+5)^5 = a^5 + 25a^4 + 250a^3 + 1250a^2+ 3125a +3125`

Explanation:

`(a+5)^5`
As per Binomial theorem,
`(a+b)^n = a^n+n*a^(n-1)*b + ((n(n-1))/2)*a^n-2*b2+ .... + b^n`
`(a+b)^n = nCn*a^n + nCn-1*a^(n-1)*b + nCn-2*a^(n-2)*b^2+ ..... +b^n`

Replacing b by 5 and n by 5,
`(a+5)^5 = a5 + 5C4*a^4*5 + 5C3*a^3*5^2 + 5C2*a^2*5^3+ 5C1*a*5^4+ 5^5`

`5C1 = 5C4 = 5, 5C2 = 5C3 =( 5*4)/(1*2) = 10`

`= a^5 + 25a^4 + 250a^3 + 1250a^2+ 3125a +3125`