Multiplying Radicals, the Smart Way

I love using smart ways to do math because it can make math so much easier. Unfortunately, many times those little tricks and methods are considered less important than the big concepts and therefore get less attention in math class than they deserve. Let’s look at how much of a difference a smart simplified sequence of steps can make to the difficulty of a problem. As an example, we will multiply radicals.

Here is the problem:

\(\sqrt{84} \cdot \sqrt{175}\)

The brute force approach is to multiply the radicands. To find the simplest form of the resulting radical, we then need to factor the radicand and find all squares that can be pulled out of the square root.

\(\sqrt{14700}\)

\(\sqrt{7 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 7}\)

\(7 \cdot 2 \cdot 5 \cdot \sqrt{3}\)

\(70 \sqrt{3}\)

Now let’s do the same problem but differently. We don’t start by multiplying but by finding the factors of both radicands right away. Then we write the product of those factors as one radicand and pull out the squares.

\(\sqrt{7 \cdot 2 \cdot 2 \cdot 3} \cdot \sqrt{5 \cdot 5 \cdot 7}\)

\(\sqrt{7 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot 5 \cdot 7}\)

\(7 \cdot 2 \cdot 5 \cdot \sqrt{3}\)

\(70 \sqrt{3}\)

The difference lies in the first step. With the second method, the product of 84 and 175, 14700, never appears. We skipped the multiplication completely and went to factoring right away instead. This does not only save us the work of multiplying large numbers but factoring 84 and 175 is also so much easier than factoring 14700!

Conclusion: Always try to find a smart way to make the math easier, and you can save a lot of time and avoid mistakes, too.