1. **State the problem:** We want to understand what $8^{1/2}$ means using the properties of integer exponents and define it consistently. 2. **Use the power of a power rule:** The rule states that $(a^m)^n = a^{m \times n}$. 3. **Apply the rule to $(8^{1/2})^2$:** $$ (8^{1/2})^2 = 8^{1/2 \times 2} = 8^1 = 8 $$ 4. **Interpretation:** For the power of a power rule to hold, raising $8^{1/2}$ to the power 2 must give 8. 5. **Which value satisfies this?** - $8$ squared is $8^2 = 64$, not 8. - $\sqrt{8}$ squared is $8$, which matches. - $\sqrt[3]{8}$ squared is $(2)^2 = 4$, not 8. - $8^2$ squared is $8^4$, not 8. - $8^3$ squared is $8^6$, not 8. 6. **Conclusion:** We define $$ 8^{1/2} = \sqrt{8} $$ because it satisfies the power of a power rule and the properties of exponents. **Final answer:** $8^{1/2} = \sqrt{8}$