Radicals – 1

Sabtu, 21 Desember 2019

Example 1 Multiply each of the following.

Solution

In all of these problems all we need to do is recall how to FOIL binomials. Recall,

(3x − 5)( x + 2) = 3x ( x) + 3x (2) − 5( x) − 5(2) = 3x² + 6x − 5x −10 = 3x² + x −10

With radicals we multiply in exactly the same manner. The main difference is that on occasionwe’ll need to do some simplification after doing the multiplication.

As noted above we did need to do a little simplification on the first term after doing the multiplication.

Don’t get excited about the fact that there are two variables here. It works the same way!

Again, notice that we combined up the terms with two radicals in them.

Not much to do with this one.

Notice that, in this case, the answer has no radicals. That will happen on occasion so don’t get excited about it when it happens.

The last part of the previous example really used the fact that

(a + b)(a − b) = a² − b²

If you don’t recall this formula we will look at it in a little more detail in the next section.

Example 2 Rationalize the denominator for each of the following.

Solution

There are really two different types of problems that we’ll be seeing here. The first two parts illustrate the first type of problem and the final two parts illustrate the second type of problem.

Both types are worked differently.

In this case we are going to make use of the fact that. We need to determine what to multiply the denominator by so that this will show up in the denominator. Once we figure this out we will multiply the numerator and denominator by this term.

Here is the work for this part.

Remember that if we multiply the denominator by a term we must also multiply the numerator by the same term. In this way we are really multiplying the term by 1 (since a/a= 1) and so aren’t changing its value in any way.

We’ll need to start this one off with first using the third property of radicals to eliminate the fraction from underneath the radical as is required for simplification.

Now, in order to get rid of the radical in the denominator we need the exponent on the x to be a 5.

This means that we need to multiply byso let’s do that.

In this case we can’t do the same thing that we did in the previous two parts. To do this one we will need to instead to make use of the fact that.

(a + b)(a − b) = a² − b²

When the denominator consists of two terms with at least one of the terms involving a radical we will do the following to get rid of the radical.

So, we took the original denominator and changed the sign on the second term and multiplied the numerator and denominator by this new term. By doing this we were able to eliminate the radical in the denominator when we then multiplied out.

This one works exactly the same as the previous example. The only difference is that both terms in the denominator now have radicals. The process is the same however.

Rationalizing the denominator may seem to have no real uses and to be honest we won’t see many uses in an Algebra class. However, if you are on a track that will take you into a Calculus class you will find that rationalizing is useful on occasion at that level.

We will close out this section with a more general version of the first property of radicals. Recall that when we first wrote down the properties of radicals we required that a be a positive number. This was done to make the work in this section a little easier. However, with the first property that doesn’t necessarily need to be the case.

Here is the property for a general a (i.e. positive or negative)

where │a│ is the absolute value of a. If you don’t recall absolute value we will cover that in detail in a section in the next chapter. All that you need to do is know at this point is that absolute value always makes a a positive number.

So, as a quick example this means that,

 

For square roots this is,

This will not be something we need to worry all that much about here, but again there are topics in courses after an Algebra course for which this is an important idea so we needed to at least acknowledge it.



Sumber

Labels: Mathematician

Thanks for reading Radicals – 1. Please share...!