This week we have begun our new unit which is Opeations on Radicals. We have started our first lesson with adding and subtracting radicals which will be my focus point in this blog. I learned this lesson and now can deal with adding and subtracting radicals with no problems along the way. I have chosen this topic because it’s what we began with and has the key points into lead us to further lessons.

One thing we need to go over before starting the lesson on how to add and subtract radicals we have to know that there a restrictions to when you CAN and CAN’T  add/subtract radicals. The rule for adding/subtracting radicals is, that we have to have the same index (root) and the same radicand (the number under the square root sign). Think about it more like collecting “Like Terms” I’m sure you have encountered that somewhere in your math journey.

Example #1:

Starting with this simple example in this example we can see that both radials have common radicands (7) and index (2) meaning we can simplify these to radicals to one radical. We get there by leaving the radicand the same and adding/subtracting coefficients. By doing that we have simplified to the simplest form.

Now, let’s try an example with subtracting and adding to see how that exactly works.

Example #2:

Her looking at the question we can see that we have a common radicand (11) in all three radicals and a common index (2). So, what we can do is keep the radicand the same subtract the usual, and then add coefficients. Look at the third radical in the question we have sqrt 11 with no coefficient, but remember the law for this space from grade 10 it’s always going to be a visible one we just don’t write you could, but not necessarily anyone could right away know a one is there.

Okay now, hopefully, you got all this information and wonder what if the radicand are not the same? My first thought when I saw this type of question was that we can’t simplify, but looking back at previous units and lessons we can recall the perfect squares and cubes list right? So let’s try that together.

Example #3:

Here the radicand is not the same therefore we have to break the radicals further (simplify radicals). We recall how to go from entire radicals to mixed radicals right? So find two numbers that could multiply to 27 which are 9 and 3 always remember one has to be a perfect square/cube or you can look at the radicals and find a common irrational square/cube for both radicals. From there we get to step there we have managed to turn into mixed fractions which is what we need and we’ve managed to get common radicands also was the main point of being able to simplify. Then just follow as usual add/subtract coefficients and keep the same radicand.

Finally, we can solve a question that includes all the steps we learned. The aim of performing addition and subtraction with radicals is to simplify the expression to its most basic form. However, contrary to what was mentioned earlier, achieving the simplest form doesn’t necessarily entail having just one radical in the answer. Occasionally, when adding or subtracting, it might not be possible to find a common radicand for all radicals, yet it’s still feasible to simplify the radicals to some extent.

Example #4:

In this instance, we applied similar steps as before and converted all the radicals into mixed radicals. During this process, not all the radicals shared a common radicand. However, two radicands in this equation matched. To attain the answer in its most simplified form, we need to consolidate like terms. Consequently, our ultimate solution comprises two mixed radicals. They cannot be subtracted due to differing radicands, and further simplification isn’t possible as there are no additional perfect squares within them.