Week 4 – Precalculus – Simplifying Expressions with Variable Radicands

Hello! This week I wanted to go over simplifying expressions with variable radicands because I feel that talking through the steps will give myself a greater understanding of the content that I am learning.

Connected to what we have learned before in Week 2, this topic relies heavily on mixed radicals and like terms.

Consider the term + – .

Notice how all of these have the base of . They have the same base, so you can add and subtract these terms easily. This term will simplify to – which will then equal 0 because those two terms have the same base.

Now that we have finished this basic question, let’s move on to a harder one.

Consider the term – + + .

Since all of these terms have the same base of x, we can simplify the expression to – . This expression will then simplify to because – have the same base.

Now let’s think: is this expression greater than or less than zero? By looking at , we can come to the conclusion that is greater than zero. So we would write the expression stating this next to our answer: x > 0.

Let’s try a harder expression.

Consider + . This expression looks intimidating, right?

Well,  look back to our knowledge of mixed radicals. We must first simplify the radicands in to a mixed radical before going any further. I usually use a factor tree to find the perfect squares that will turn this expression in to a mixed radical.

After our simplification, the expression should look like this:  + .

To check your answer, you can turn this expression back to an entire radical. If it matches up with the original expression, you have the correct simplification.

Notice how the bases are the same. From here, we can add these two terms together and it will give us our final answer: .

I hope that this helps!