1. **Problem statement:** Evaluate and simplify numerical expressions raised to fractional indices. 2. **Formula and rules:** A fractional exponent $a^{\frac{m}{n}}$ means the $n$th root of $a$ raised to the $m$th power: $$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$ Important rules: - The base $a$ must be non-negative if $n$ is even. - Simplify roots and powers step-by-step. 3. **Example:** Simplify $16^{\frac{3}{4}}$. 4. **Step 1:** Express as root and power: $$16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3$$ 5. **Step 2:** Calculate the 4th root of 16: $$\sqrt[4]{16} = 2$$ because $2^4 = 16$. 6. **Step 3:** Raise the result to the 3rd power: $$2^3 = 8$$ 7. **Answer:** $$16^{\frac{3}{4}} = 8$$ This method applies to any numerical expression with fractional exponents: first find the root, then raise to the power.