GED Math: Exponents on Whole Numbers

Welcome to another video from the Ultimate GED Math Course. In this video we will continue our lesson on operations on whole numbers. We will be looking at exponents in todays video.

There are a lot more to exponents than what we will cover in this video. This is because we will cover exponents again when we deal with and extensively when we look at exponents in Algebra.

Before we even look at our question I want us to solve this problem permanently 

Exponents shows the number of times a number multiplies itself.

2³ is not the same as 2 times 3.

2³ means 2 multiplying itself 3 times. That is 2 times 2 times 2. This is 2 times 2 which is 4. Then 4 times 2, which is 8. This is not the same as 2 times 3, which is simply 6.

5⁴ is 5 x 5 x 5 x 5. This will give 625. It is not 5 times 4. Which is simply 20.

I cannot count the number of times I’ve seen students get this part wrong. Please take note of it if you’re not already familiar with it.

Now that I’ve gotten this out of the way, Let’s look at our question.

Question 5

We are supposed to calculate 3^10 times 3^5 all over 3^11, without using a calculator.

If you have two numbers multiplying and the bases are the same you can just add the exponent.

Example let’s look at 2^3 times 2^4

Because they both have the same bases, which is 2 in this case. We can just add the exponents.

So this will be 2 exponent 3+4, which is 2^7

Now if you have a number being divided by another number with the same base, then you can simply subtract the exponents.

So 2^5 ÷ 2^2 will be 5 exponent 5-2, which is 5^3.

Please these two rules applies only when the bases are the same. It will not work for 3^5 times 2^4. The bases are different. One is 3 and the other is 2. 

Even if the exponents are the same and the bases are different it won’t work.

Example 3^5 times 2^5. The exponents being the same is irrelevant. The bases are not the same so we can’t use this.

So for our question, we can see that they all have the base of 3 so the rules can apply.

So 3^10 is multiplying 3^5, so we can add the exponents to get 3 exponent 10+5. This will give as 3^15.

This 3^15 is being divided by 3^12.

We know that when terms with the same base divide, we can simply subtract their exponent.

So we have 3 exponent 15-12, which is 3^3.

So this is the answer.

They could have required you to further simplify the 3^3. 

Which will be 3 times 3 times 3.

This will be 27.

Before we look at our next question let’s explore the concept of exponent 1 and exponent zero. We will explain them fully but you don’t really need the explanation to solve GED Math questions. If you can remember the general statements you’re good. The explanations are for those who care to know.

The Concept of exponent 1

Any number exponent 1 is that same number. 

Example 5^1 is 5

That’s all you need to remember if you don’t care about why.

We said exponents represents the number of times a number repeats itself.

So if we say, 2 exponent 1, then it means 2 repeats just one. Which will simply be 2.

So 76 exponent 1 is 76.

The Concept of exponent zero.

Any number exponent zero is 1. 

Example

5 exponent 0 is 1.

That’s all you need to remember if you don’t care about why.

Let’s look at 2^3 ÷ 2^3.

In mathematics any number divided by itself is 1.

So 2 ÷ 2 is 1.

100 ÷ 100 is 1.

Basically if the numerator and denominator of a fraction are the same then the value is 1.

The numerator is basically the top value and the denominator is basically the bottom value.

So we know the value of this is 1, since the numerator and denominator are the same.

But in exponents, we learned that when two things divide, provided the base is the same, you can subtract the exponent. We learned this from the previous question.

So this can be written as 2 exponent 3 minus 3.

3 minus 3 is zero.

So this is 2 exponent zero.

2^3 over 2^3 is 1 and 2^3 over 2^3 is also 2^0.

So we can say therefore that 2^0 must be equal to 1.

We can take any number and exponent, so far as the numerator and denominator are the same we will end up with the same thing.

Here we chose 5^4 and we ended up with 5^0 = 1

Here we chose 7^2 and we ended up with 7^0 = 1

Question 6.

Calculate 2^-3 x 16. (Do NOT use a calculator)

This was the first question on our GED Math 2021 video and some students weren’t able to solve it, so let’s look at it here.

Method 1.

I don’t really expect most people at this stage in the GED Course to use this method but I want to throw it out there because it’s easy and straightforward.  Method 2 is what most of you will be familiar with.

This first method just requires that you know that 16 can be written as 2^4. Try it on your calculator. Once you know that, then you can replace the 16 with 2^4.

We have 2^-3 x 2^4.

We learned earlier that if you have two numbers multiplying and the bases are the same you can just add the exponent.

So here we will have 2 exponent -3 + 4.

We haven’t done negative yet but -3 + 4 is the same as 4 minus 3. Which is 1

So we have 2 exponent 1.

We know that any number exponent 1 is the same number, so 2 exponent 1 is simply 2. This is our answer.

Method 2

First we have to know that we can move a number with an exponent from the numerator to the denominator (or denominator to numerator) by changing the sign of the exponent.

Example: 

If you have 2 times 7^-4 over 3, We can move the 7^-4 from the numerator to the denominator. If we do that then we have to change the exponent -4 to exponent positive 4.

So we have 2 over 3 times 7 exponent positive 4.

The 7^-4 in the numerator became 7 exponent positive 4 in the denominator.

Notice that there’s this 2 in the numerator and 3 in the denominator. If you are not given any values in either one or both, you can use 1. This is not a necessary step but can serve as a guide. If you want more explanation on why we can put 1, you can ask in the comment section or visit ultimateGed.com for more help.

Let’s go to our question.

We have 2^-3 times 16.

Here we do not have a denominator, so we can use 1. So we’ll have 2^-3 times 16 over 1.

We know we can move the 2^-3 to the denominator to become  2^3. Notice that the exponent -3 became exponent positive 3.

1 times 2^3 is simply 2^3.

So we have 16 over 2^3.

2^3 is 2 x 2 x 2 which is 8.

We have 16 divided by 8.

This will give us 2 as our final answer.

Please note that for teaching purposes we expand and explain answers in details, on your GED test please don’t waste time on writing and expanding anything.

We will end this video here. Please like and share this video and subscribe for more. You can also visit UltimateGED.com, we will be adding more content there.

Have a great day, see you in the next video.