1. Without using calculators or other computational tools, determine which of the two numbers is greater:

$$
\frac{1+\sqrt{5}}{2} \text { or } \sqrt[6]{18} ?
$$
1. Answer: $\sqrt[6]{18}$.

Let's raise both numbers to the sixth power. To find $\left(\frac{1+\sqrt{5}}{2}\right)^{6}$, we can use formulas for simplified multiplication, sequentially cubing and squaring. However, let's consider a simpler method from a computational standpoint.

First, note that the number $x=\frac{1+\sqrt{5}}{2}$ is a root of a quadratic equation, from which $x^{2}=x+1$. Then $x^{3}=x^{2}+x=2 x+1$, and we can sequentially express the other powers: $x^{4}=2 x^{2}+x=3 x+2, x^{5}=3 x^{2}+2 x=5 x+3$, $x^{6}=5 x^{2}+3 x=8 x+5$. Now it is clear that $\left(\frac{1+\sqrt{5}}{2}\right)^{6}=4(1+\sqrt{5})+5=\sqrt{80}+9<$ $9+9=18$, so $\frac{1+\sqrt{5}}{2}<\sqrt[6]{18}$.