1. **State the problem:** Simplify the expression $$\sqrt{45} - \frac{1}{\sqrt{5}}$$. 2. **Recall the formulas and rules:** - Simplify square roots by factoring out perfect squares: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. - Rationalize denominators to remove square roots from the denominator: multiply numerator and denominator by the conjugate or the root itself. 3. **Simplify $$\sqrt{45}$$:** $$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$$. 4. **Rationalize the denominator of $$\frac{1}{\sqrt{5}}$$:** Multiply numerator and denominator by $$\sqrt{5}$$: $$\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}$$. 5. **Rewrite the expression:** $$3\sqrt{5} - \frac{\sqrt{5}}{5}$$. 6. **Combine like terms:** Express $$3\sqrt{5}$$ as $$\frac{15\sqrt{5}}{5}$$ to have a common denominator: $$\frac{15\sqrt{5}}{5} - \frac{\sqrt{5}}{5} = \frac{15\sqrt{5} - \sqrt{5}}{5} = \frac{14\sqrt{5}}{5}$$. **Final answer:** $$\frac{14\sqrt{5}}{5}$$.