1. **State the problem:** Simplify and understand the expressions involving $t$ and square roots, and evaluate the product $\sqrt{5} \times \sqrt{3} \times \sqrt{2}$.\n\n2. **Recall the rules:**\n- $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$ for positive $a,b$.\n- $t^{1/2} = \sqrt{t}$.\n- Division and multiplication of powers with the same base: $t^m / t^n = t^{m-n}$ and $t^m \times t^n = t^{m+n}$.\n\n3. Simplify $\frac{t}{\sqrt{t}}$:\n$$\frac{t}{\sqrt{t}} = \frac{t}{t^{1/2}} = t^{1 - \frac{1}{2}} = t^{\frac{1}{2}} = \sqrt{t}$$\n\n4. Simplify $\frac{\sqrt{t}}{t}$:\n$$\frac{\sqrt{t}}{t} = \frac{t^{1/2}}{t^1} = t^{\frac{1}{2} - 1} = t^{-\frac{1}{2}} = \frac{1}{t^{1/2}} = \frac{1}{\sqrt{t}}$$\n\n5. Simplify $\frac{t}{t \sqrt{t}}$:\n$$\frac{t}{t \sqrt{t}} = \frac{t}{t \times t^{1/2}} = \frac{t}{t^{1 + \frac{1}{2}}} = \frac{t}{t^{\frac{3}{2}}} = t^{1 - \frac{3}{2}} = t^{-\frac{1}{2}} = \frac{1}{\sqrt{t}}$$\n\n6. Simplify $\sqrt{t} \times \sqrt{t}$:\n$$\sqrt{t} \times \sqrt{t} = t^{1/2} \times t^{1/2} = t^{\frac{1}{2} + \frac{1}{2}} = t^1 = t$$\n\n7. Evaluate $\sqrt{5} \times \sqrt{3} \times \sqrt{2}$:\n$$\sqrt{5} \times \sqrt{3} \times \sqrt{2} = \sqrt{5 \times 3 \times 2} = \sqrt{30}$$\n\n8. Simplify $t \times \sqrt{t}$:\n$$t \times \sqrt{t} = t^1 \times t^{1/2} = t^{1 + \frac{1}{2}} = t^{\frac{3}{2}} = t^{1.5}$$\n\n**Final answers:**\n- $\frac{t}{\sqrt{t}} = \sqrt{t}$\n- $\frac{\sqrt{t}}{t} = \frac{1}{\sqrt{t}}$\n- $\frac{t}{t \sqrt{t}} = \frac{1}{\sqrt{t}}$\n- $\sqrt{t} \times \sqrt{t} = t$\n- $\sqrt{5} \times \sqrt{3} \times \sqrt{2} = \sqrt{30}$\n- $t \times \sqrt{t} = t^{3/2}$