Summary of the index laws

1. To multiply powers of the same base, add the indices.
   
 \(a^m × a^n = a^{m+n}\)
2. To divide powers of the same base, subtract the indices.
   
 \(a^m ÷ a^n = a^{m-n}\)
3. To raise a power to a power, multiply the indices.
   
 \((a^m)n = a^{mn}\)
4. The power of a product is the product of the powers.
   
 \((ab)^n = a^nb^n\)
5. The power of a quotient is the quotient of the powers.
   \((\dfrac{a}{b})^n = \dfrac{a^n}{b^n}\)



Zero index

\begin{align}\hspace{-6mm}\text{Clearly,}\ \dfrac{4^3}{4^3} &= 1.\ \text{If we apply the second index law, then we have}\\\\ \dfrac{4^3}{4^3} &= 4^{3–3} = 4^0 = 1.\end{align}

Hence we define a\(\bf ^0 = 1\) for all non-zero a.

So \(2^0 = 1, 2 × 15^0 = 2 × 1 = 2\ \text{and}\ (2 × 15)^0 = 1.\)