1. **State the problem:** We need to calculate $8 \div 2^{4/5}$ and express the answer as a mixed number in its lowest terms. 2. **Recall the rules:** Division by an exponent means dividing by the number raised to that power. We can write this as: $$8 \div 2^{4/5} = \frac{8}{2^{4/5}}$$ 3. **Rewrite 8 as a power of 2:** Since $8 = 2^3$, substitute: $$\frac{2^3}{2^{4/5}}$$ 4. **Use the quotient rule for exponents:** $$\frac{a^m}{a^n} = a^{m-n}$$ So, $$2^{3 - \frac{4}{5}} = 2^{\frac{15}{5} - \frac{4}{5}} = 2^{\frac{11}{5}}$$ 5. **Convert the exponent to a mixed number:** $$\frac{11}{5} = 2 + \frac{1}{5}$$ 6. **Rewrite the expression:** $$2^{2 + \frac{1}{5}} = 2^2 \times 2^{\frac{1}{5}} = 4 \times 2^{\frac{1}{5}}$$ 7. **Express $2^{1/5}$ as a root:** $$2^{\frac{1}{5}} = \sqrt[5]{2}$$ 8. **Final answer as a mixed number:** $$4 \sqrt[5]{2}$$ This is a mixed number with integer part 4 and fractional part $\sqrt[5]{2}$ in simplest form.