1. **State the problem:** Simplify and analyze the function $$r(z) = z^{-8} - z^{\frac{1}{2}}$$. 2. **Recall the rules:** - Negative exponents mean reciprocal: $$z^{-8} = \frac{1}{z^8}$$. - Fractional exponents represent roots: $$z^{\frac{1}{2}} = \sqrt{z}$$. 3. **Rewrite the function using these rules:** $$r(z) = \frac{1}{z^8} - \sqrt{z}$$ 4. **Domain considerations:** - $$z^8$$ is defined for all $$z \neq 0$$ (denominator cannot be zero). - $$\sqrt{z}$$ requires $$z \geq 0$$ if considering real numbers. 5. **Final simplified form:** $$r(z) = \frac{1}{z^8} - \sqrt{z}$$ This is the simplified expression showing the function in terms of radicals and fractions. **Answer:** $$r(z) = \frac{1}{z^8} - \sqrt{z}$$