1. **State the problem:** Simplify the expression $a\sqrt{a^{-1}} \cdot \sqrt{a^{-1}} + \sqrt{a^{-1}}$. 2. **Recall the rules:** - $\sqrt{x} = x^{\frac{1}{2}}$. - When multiplying powers with the same base, add exponents: $x^m \cdot x^n = x^{m+n}$. - When multiplying square roots, $\sqrt{x} \cdot \sqrt{y} = \sqrt{xy}$. 3. **Rewrite the expression using exponents:** $$a \cdot (a^{-1})^{\frac{1}{2}} \cdot (a^{-1})^{\frac{1}{2}} + (a^{-1})^{\frac{1}{2}}$$ 4. **Simplify the powers inside the product:** $$(a^{-1})^{\frac{1}{2}} \cdot (a^{-1})^{\frac{1}{2}} = a^{-\frac{1}{2}} \cdot a^{-\frac{1}{2}} = a^{-1}$$ 5. **Substitute back:** $$a \cdot a^{-1} + a^{-\frac{1}{2}}$$ 6. **Simplify $a \cdot a^{-1}$:** $$a^{1} \cdot a^{-1} = a^{1-1} = a^{0} = 1$$ 7. **Final expression:** $$1 + a^{-\frac{1}{2}} = 1 + \frac{1}{\sqrt{a}}$$ **Answer:** $1 + \frac{1}{\sqrt{a}}$