Zero Index

Any base that has a power of zero has a value of one. it does not matter whether the base is a number or a pronumeral. If the power of zero; its value is one. We can show this by looking at the following example which can be simplified using two different methods.

$\textbf{Method 1}$ $$ \begin{align} \displaystyle 5^4 \div 5^4 &= \dfrac{5 \times 5 \times 5 \times 5}{5 \times 5 \times 5 \times 5} \\ &= \dfrac{625}{625} \\ &= 1 \\ \end{align} $$ $\textbf{Method 2}$ $$ \begin{align} \displaystyle 5^4 \div 5^4 &= \dfrac{5^4}{5^4} \\ &= 5^{4-4} \\ &= 5^0 \\ \end{align} $$ Since the two results should be the same, $5^0$ must equal to $1$.
Any base that has an index (power) of zero is equal to $1$. $$a^0 = 1$$

Example 1

Find the value of $12^0$.

Example 2

Find the value of $(15a^4b^5c^6)^0$.

Example 3

Find the value of $5^0 + 5$.

Example 4

Simplify $4a^2b^0$.

Example 5

Simplify $\dfrac{9x^4 \times 4x^7}{2x^6 \times 3x^5}$.