As an inventory specialist, you need to determine the side length of a perfectly cubic storage bin that holds 300 cubic feet of material. The exact side length is $\sqrt[3]{300}$ feet. Using the method of estimating higher roots by finding consecutive perfect powers, between which two whole numbers does the side length of this bin lie?

A logistics coordinator needs to estimate the edge length of a cubic storage container that has a volume of ${}50$ cubic feet. The exact edge length is $\sqrt[3]{50}$ feet. Arrange the steps the coordinator should take to estimate this value between two consecutive whole numbers using the bounding method.

An engineer is estimating a dimension calculated as $\sqrt[4]{100}$ centimeters. To bound this value between two consecutive whole numbers, the engineer identifies that ${}3^4 = 81$ and ${}4^4 = 256$. Consequently, the engineer concludes that $\sqrt[4]{100}$ must lie between 3 and 4.

Logistics: Estimating Cubic Box Dimensions

During a site assessment, you need to quickly estimate an $n$th root for a spatial calculation without using a calculator. To determine the consecutive whole numbers the root lies between, you must locate the ____ between two consecutive perfect $n$th powers.