Question

How do you use the Binomial Theorem to find the value of `999^3`?

Answer 1

Rewrite `999^3` as `(10^3-1)^3` and apply the binomial theorem to find that

`999^3=997002999`

Explanation:

The binomial theorem states that `(a+b)^n = sum_(k=0)^n((n),(k))a^(n-k)b^k`

where `((n),(k)) = (n!)/(k!(n-k)!)`

For this problem, we will only need to calculate for `n=3`, and we will find that `((3),(0))=((3),(3))=1` and `((3),(1))=((3),(2))=3`
(Try verifying this)

Noting that it is much easier to calculate powers of `10` and `1` compared to `999`, we can rewrite `999` as `1000-1=10^3-1` and apply the binomial theorem:

`999^3 = (10^3-1)^3`

`=((3),(0))(10^3)^3+((3),(1))(10^3)^2(-1)+((3),(2))(10^3)(-1)^2+((3),(3))(-1)^3`

`=10^9-3*10^6+3*10^3-1`

`=1000000000-3000000+3000-1`

`=997002999`