1. **Stating the problem:** We want to understand the properties of rational exponents and see some examples. 2. **Definition:** A rational exponent is an exponent that is a fraction, written as $\frac{m}{n}$ where $m$ and $n$ are integers and $n>0$. 3. **Key property:** For any positive number $a$ and integers $m,n$ with $n>0$, the rational exponent is defined as: $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$ This means raising $a$ to the $\frac{m}{n}$ power is the same as taking the $n$th root of $a$ raised to the $m$th power. 4. **Important rules of rational exponents:** - $a^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{m}{n} + \frac{p}{q}}$ - $\left(a^{\frac{m}{n}}\right)^k = a^{\frac{m}{n} \cdot k}$ - $\left(ab\right)^{\frac{m}{n}} = a^{\frac{m}{n}} b^{\frac{m}{n}}$ - $\frac{a^{\frac{m}{n}}}{b^{\frac{m}{n}}} = \left(\frac{a}{b}\right)^{\frac{m}{n}}$ 5. **Example 1:** Simplify $8^{\frac{2}{3}}$ $$8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$$ 6. **Example 2:** Simplify $16^{\frac{3}{4}}$ $$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$$ 7. **Example 3:** Simplify $\left(27^{\frac{1}{3}}\right)^2$ $$\left(27^{\frac{1}{3}}\right)^2 = 27^{\frac{1}{3} \cdot 2} = 27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$$ These properties allow us to work with roots and powers in a unified way using rational exponents.