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Radicals are often a source of mild confusion mechanically, if not conceptually. However, it is worth giving at least a cursory motivation as to how radicals appear in practice, which will also motivate our alternate method of writing radicals (as powers) in the next topic.

Before we dive in, it helps to establish a few key pieces of vocabulary. Notice that “radical” can be used to reference either just the symbol that encompasses the radical expression, or it can mean an entire expression that contains a radical - which one it means should be made clear by context.

As we will see, there is a fundamental difference between even radicals, like square or fourth roots, and odd radicals, like cube or fifth roots. For this reason, it’s handy to have a term to reference what “power” the root is using.

Finally, we will often need to discuss the content of the radical expression that is within the radical symbol, so it helps to have a term for this as well.

Where do we find a radical in the wild?

A radical is most often found in practice by trying to isolate some term in an equality that is being raised to a power. In fact, the most common way to encounter a radical is when we know some end-result information we want, but need to determine some initial/prior step/information, ie when we are trying to “work backward” from the end of some process back to a previous step. This is often the case because we are usually working in the real-world of three dimensional space and powers occur rather naturally in the geometry of the real world. ^[1] (In fact, this is especially true because most optimal conditions tend to involve finding spaces that maximize or minimize a dimensional quantity relative to another. For example you may want to “maximize volume while minimizing surface area” to minimize costs to build a container. It turns out that, without other constraints, almost always the answers to these problems are “regular” polygons or perfectly round shapes, e.g. squares, cubes or spheres, which involve a single dimension raised to a power. Like how a cube has volume $w^3$ and each face has area $w^2$ for example.) Let’s see an example of the kind of problem we aim to solve by the end of this topic.

Note: the example below contains most of the things we aim to learn, but not all, which means it’s ok if you don’t follow every step yet. Make sure to return to this example at the end of the topic, at which point this should seem like an ‘obviously easy’ problem.