Rational Exponents

Rabu, 22 Januari 2020

Now that we have looked at integer exponents we need to start looking at more complicated exponents. In this section we are going to be looking at rational exponents. That is exponents in the form,

where both m and n are integers.

We will start simple by looking at the following special case,

where n is an integer. Once we have this figured out the more general case given above will actually be pretty easy to deal with.

Let’s first define just what we mean by exponents of this form.

In other words, when evaluatingwe are really asking what number (in this case a) did we raise to the n to get b. Oftenis called the n^th root of b.

As the last two parts of the previous example has once again shown, we really need to be careful with parenthesis. In this case parenthesis makes the difference between being able to get an answer or not.

Also, don’t be worried if you didn’t know some of these powers off the top of your head. They are usually fairly simple to determine if you don’t know them right away. For instance in the part b we needed to determine what number raised to the 5 will give 32. If you can’t see the power right off the top of your head simply start taking powers until you find the correct one. In other words compute 2⁵ , 3⁵ , 4⁵ until you reach the correct value. Of course in this case we wouldn’t need to go past the first computation.

The next thing that we should acknowledge is that all of the properties for exponents that we gave in the previous section are still valid for all rational exponents. This includes the more general rational exponent that we haven’t looked at yet.

Now that we know that the properties are still valid we can see how to deal with the more general rational exponent. There are in fact two different ways of dealing with them as we’ll see. Both methods involve using property 2 from the previous section. For reference purposes this property is,

(a^{n) m} = a^nm

So, let’s see how to deal with a general rational exponent. We will first rewrite the exponent as follows.

In other words we can think of the exponent as a product of two numbers. Now we will use the exponent property shown above. However, we will be using it in the opposite direction than what we did in the previous section. Also, there are two ways to do it. Here they are,

Using either of these forms we can now evaluate some more complicated expressions.

We will leave this section with a warning about a common mistake that students make in regards to negative exponents and rational exponents. Be careful not to confuse the two as they are totally separate topics.

In other words,

and NOT

This is a very common mistake when students first learn exponent rules.

Sumber

Labels: Mathematician

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