1. **Problem Statement:** Evaluate expressions involving rational exponents and radicals such as $16^{3/4}$, $16^{-3/4}$, $8^{2/3}$, etc. 2. **Formula and Rules:** - Rational exponents: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ - Negative exponents: $a^{-k} = \frac{1}{a^k}$ - When evaluating, simplify inside the root first if possible. - For negative bases with fractional exponents, be careful: if the root is even, the result may be complex or undefined in reals. 3. **Example: Evaluate $16^{3/4}$** - Using the formula: $16^{3/4} = \sqrt[4]{16^3} = (\sqrt[4]{16})^3$ - Since $\sqrt[4]{16} = 2$ (because $2^4=16$), then - $16^{3/4} = 2^3 = 8$ 4. **Example: Evaluate $16^{-3/4}$** - Use negative exponent rule: $16^{-3/4} = \frac{1}{16^{3/4}}$ - From above, $16^{3/4} = 8$ - So, $16^{-3/4} = \frac{1}{8}$ 5. **Example: Evaluate $8^{2/3}$** - $8^{2/3} = \sqrt[3]{8^2} = (\sqrt[3]{8})^2$ - $\sqrt[3]{8} = 2$ (since $2^3=8$) - So, $8^{2/3} = 2^2 = 4$ 6. **Example: Evaluate $8^{-2/3}$** - $8^{-2/3} = \frac{1}{8^{2/3}}$ - From above, $8^{2/3} = 4$ - So, $8^{-2/3} = \frac{1}{4}$ 7. **General Steps:** - Convert the rational exponent to a root and power. - Simplify the root if possible. - Apply the power. - If exponent is negative, take reciprocal. This method applies to all similar problems in the list. **Final note:** For negative bases with fractional exponents, if the denominator of the exponent is even, the expression is not a real number. For odd denominators, you can take the root of the absolute value and then apply the sign accordingly.