problem: Paula invests $\$10,\!000$ at the start of a 5 year period at an interest rate of $10\%$. At the end of those 5 years, how much is her investment worth if the interest is compounded quarterly?  Express your answer rounded to the nearest cent.
solution: In the first quarter, Paula earns $\frac{0.10}{4}(\$10,\!000)$ in interest, so her investment is worth $\$10,\!000 +\frac{0.10}{4}(\$10,\!000) = \left(1 + \frac{0.10}{4}\right)(\$10,\!000)$. Similarly, the value of her investment is multiplied by $1 + \frac{0.10}{4}$ each quarter, so after 5 years, which is $5\cdot 4 = 20$ quarters, her investment is worth \[\left(1 + \frac{0.10}{4}\right)^{5\cdot 4}(\$10,\!000) \approx \boxed{\$16,\!386.16}.\]