Index laws

The following index laws have been established, where a and b are integers and m and  n are non-zero whole numbers.

1. To multiply powers with the same base, add the indices.
   \(a^m × a^n = a^{m+n}\)
2. To divide powers with the same base, subtract the indices.
   \(\dfrac{a^m}{a^n}= a^{m–n}\), provided \(m > n\) and \(a \neq 0\)
3. To raise a power to a power, multiply the indices.
   \((a^m)^n = a^{mn}\)
4. A power of a product is the product of the powers.
   \((ab)^m = a^m b^m\)
5. A power of a quotient is the quotient of the powers.
   \((\dfrac{a}{b})^m = \dfrac{a^m}{b^m}= a^{m–n}\), provided \(b \neq 0\)

These laws also hold when a and b are real.

Example 2

Write \(5^3 × 5^6\) as a single power of 5.

Solution

\begin{align}5^3 × 5^6 &= 5^{3+6}\\ &= 5^9\end{align}

Example 3

Simplify \(\dfrac{3^5}{3^2}\)

Solution

\begin{align}\dfrac{3^5}{3^2} &= 3^{5-2}\\ &=3^3\\ &=27\end{align}

Example 4

Simplify \((\dfrac{x^3}{y^2})^2 × (\dfrac{y}{x})^4\)

Solution

\begin{align}(\dfrac{x^3}{y^2})^2 × (\dfrac{y}{x})^4&= \dfrac{x^6}{y^4}×\dfrac{y^4}{x^4}\\                        &= x^2\end{align}