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