Laws of Exponents (Index Laws)

$\textbf{Laws of Exponents (Index Laws)}$

$$\large a^x \times a^y = a^{x+y}$$
To $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base and $\textit{add}$ the exponents.

$$\large \dfrac{a^x}{a^y} = a^x \div a^y = a^{x-y}$$
To $\textit{divide}$ numbers with the $\textit{same base}$, keep the base and $\textit{substract}$ the exponents.

$$\large (a^x)^y = a^{x \times y}$$
When $\textit{raising a power to a power}$, keep the base and $\textit{multiply}$ the exponents.

$$\large (ab)^x = a^xb^x$$
The power of a product is the product of the powers.

$$\large \Big(\dfrac{a}{b}\Big)^x = \dfrac{a^x}{b^y}, b \ne 0$$
The power of a quotient is the quotient of the powers.

$$\large a^0 = 1, a \ne 0$$
Any non-zero number raised to the power of zero is 1.

$$\large a^{-x} = \frac{1}{a^x} \text{ and } \frac{1}{a^{-x}} = a^x, a \ne 0$$

Example 1

Simplify $a^5 \times a^6$ using Laws of Exponents (Index Laws).

\begin{align} \displaystyle a^5 \times a^6 &= a^{5+6} \\ &= a^{11} \end{align}

Example 2

Simplify $\dfrac{a^7}{a^3}$.

\begin{align} \displaystyle \dfrac{a^7}{a^3} &= a^{7-3} \\ &= a^4 \end{align}

Example 3

Simplify $(a^3)^4$.

\begin{align} \displaystyle (a^3)^4 &= a^{3 \times 4} \\ &= a^{12} \end{align}

Example 4

Simplify $(a^2b^3)^4$.

\begin{align} \displaystyle (a^2b^3)^4 &= a^{2 \times 4} b^{3 \times 4} \\ &= a^8 b^{12} \end{align}

Example 5

Write $\dfrac{a^{-2}}{b^{-3}}$ without negative exponents.

\begin{align} \displaystyle \dfrac{a^{-2}}{b^{-3}} &= \dfrac{b^3}{a^2} \end{align}