The zero index

Clearly, \(\dfrac{5^3}{5^3}\) = 1. On the other hand, applying index law 2, ignoring the condition m > n, we have = \(5^0\). If the index laws are to be applied in this situation, then we need to define \(5^0\) to be 1.
More generally, if \(a\not=0\) then we define \(a^0\) = 1.
Note that \(0^0\) is not defined. It is sometimes called an indeterminant form.

Example 5

Simplify \((2xy^2)^0 × (3x^2y)^3\)

Solution

\begin{align}(2xy^2)^0 × (3x^2y)^3 &= 1 × 3^3x^6y^3\\                                   &= 27x^6y^3\end{align}