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.

Welcome to Chapter 2 of the ultimate GED Math Course from UltimateGED.com. In this Chapter, we will be looking at Operations on  Whole Numbers. We will be looking at Addition and Subtraction in this video and we will look at other operations like exponent, multiplication and division in subsequent videos. 

This video is going to be brief since most students are familiar with this topic. We have a full video  with about 10 examples on addition and subtraction of whole numbers from our pre-algebra course. We’ll put a link in the description for those who need extra help

Question 3

John sold $345 worth of goods on Monday, $843 worth of goods on Wednesday and $1524 of goods on Friday. How much money did he receive on those three days. (Do NOT use a Calculator).

You are usually not allowed to use a calculator on GED Math questions involving operations on whole numbers.

This question is asking for the total amount of money. You will therefore have to add the three amounts. The work involved in adding whole numbers is to align the numbers from the unit column(or the right). Add them starting form the unit column and carry values to the next column if you have 2 digits values.

So here we have 345, then 843, and then 1524. Please make sure they are aligned starting from the unit column( that’s the right). It cannot look like any of these. All three numbers must align at the right.

Let’s add. Starting from the right. 

5 + 3 + 4 is 12. Because it’s a two digit number, we will put the 2 here and carry the 1 to the next column. 

We move to the next column. We have 1 + 4 + 4 + 2. This is 11. Again this is a two digit number. We will keep the 1 and carry 1 to the next column. 

We move to the next column. We have 1 + 3 + 8 + 5. This is 17. We have a two digit number. We will keep the 7 and carry the 1 to the next column.

Finally, we have 1 + 1 here. This is 2.

So our final answer is $2,712

Question 4.

A business woman made $34,937 in sales. She then pays $3,556 in taxes on that money. How much money is left. (Do NOT use a Calculator).

This is a typical subtraction question. Here you are supposed to subtract the $3556 from $34937. Like we did in addition, you’ll have to align the values from the unit column. That’s aligning from the right.

So we have 34937 minus 3556. 

We start from the units column. 7 minus 6 is 1. 

We move to the next column. we have 3 minus 5. Since 3 is less than 5, we will have to borrow 1 from the next column. 

This is 9, when we borrow 1, it now becomes 8. When you borrow, the value is 10, so we will add 10 plus 3 to get 13. For simplicity sake, we will say that when you borrow, you’ll just put 1 in front of your number. So we have 13 minus 5, which is 8.

Next we know we have 8 here now. 8 minus 5 is 3.

We move to the next column. 4 minus 3 is 1.

Finally, we have 3 here. Nothing to subtract from, so we have 3 minus zero which is 3.

So the amount of money left is $31381.

We will end this video here. If you need more help, check out the link in the description to get the full Pre-Algebra video on Addition and Subtraction of whole numbers.

Please like, share, subscribe and turn on your notification. It’s extremely important you watch our next video on exponents and all subsequent videos.

Thank You, have a great day. See you in the next video.

Welcome to the complete Ultimate GED Math Course from UltimateGED.com.  The purpose of this course is to start you from scratch and give you everything you need to pass the GED Test with EASE. We’ve helped a lot of people pass and we are 100% sure we can help you Pass. All you need to do is watch all the videos in this course and be engaged in the comment section. Ask questions and answer questions. We are going to cover everything. For every topic we cover here, we will put a link in the description to ultimateged.com, where you can get more examples. Ok Let’s dive in. We are starting with Introduction to numbers.

Question 1. The Number 2 is …

Here we are supposed to select all that applies. To get this right. We need to know the kind of numbers. Counting or natural numbers are the numbers we count, so we have 1, 2, 3, 4 and so on. When you add zero to these numbers, we have whole numbers. So whole numbers are 0, 1, 2, 3, 4 and so on. When we add negatives, we have integers. We have  0, 1, 2,3 and so on and we can add, -1, -2, -3 and so on to get our integers. When you add fractions to integers, then you have rational numbers. So we have  -2, -3/2, -1, 0, 1/2, 1, 5/4, 2 and so on are rational numbers. We’ll take a deeper dive into rational numbers with the next question. Let’s just use this information for now. Ok, so looking at this, you’ll notice that once a number is in a lower level, it means it’s in all the levels above it. If a number is here. Then it will automatically be here and here. For our question, we can say that 2 is a counting or natural number, check. 2 is a whole number, check.2 is an integer, check and 2 is a rational number, check. So we have our answer

Let’s try -5. See if you can do it yourself.

-5 is not a natural number because natural numbers are 1, 2, 3 and so on. -5 is not a whole number, because whole numbers are 0, 1, 2, 3 and so on. -5 is and integer. We know that negative counting numbers are integers. And -5 is a rational number. Let’s try a more trick question. If x is a positive integer, then x is also… Please try this and leave your answer in the comment section. You can visit UltimateGED.com for the solution and more examples if you are interested.

Let’s look at question 2.

How many elements of the set {0,-1, π, -4/7, √2,} are rational numbers. A Rational number is the ratio of two integers. We know our integers are positive counting numbers (1, 2, 3, and so on) and negative counting numbers  (-1, -2, -3 and so on) and zero. So if we take any of the numbers over another number we have a rational number. So -2 over 3 is rational. 5 over 7 is also rational So 4 over 1 is also rational. Note that any number over 1 is the same number. So 4 over 1 can simple be written as 4. Please note that for a rational number the denominator or number at the bottom CANNOT be zero. Although we said you can pick any two integers. So, example, 8 over 0 is not a rational number. It is undefined. Please NOTE this as the exception. But the top or numerator can be zero. So zero over 8 is a rational number. I’ll post a video on UltimateGed.com for those who want to learn more on the concept of 1 and zero in math. It’s definitely something that will help you in math. Moving on, Irrational basically means NOT rational. You cannot write it as a ratio of two integers. The most common irrational numbers you’ll find on the GED are π, √2 and √3. Let’s look at our question now So 0 is a rational number. We know we can write 0 as 0/8 or 0/1 or 0/1000. 0 over any number is zero. -1 is also a rational number. We can write it as -1/1. π is an irrational number as we just learnt here. We will explain that more in later lessons but please take note of it. -4/7 is a rational number. It is an integer -4 over another integer 7. √2 is an irrational number. Please don’t assume all roots are irrational numbers. Example √9 is a rational number. Because the square root of 9 is 3. So we can say that 3 of the elements are rational. When dealing with fractions or integers, it’s easy to tell that a number is rational, but when dealing with decimals it becomes quite confusing if you don’t know what you’re looking for. 

In this question we are supposed to find the number that is irrational.

For decimals, we look at an irrational number as a number that has non-repeating and non-terminating decimal. Therefore for a Rational number the decimals must either repeat. Like 0.22222 repeating. We use this dots to show that the number continues. Here the 2 repeats or the decimals must terminate. Example 7.35. there are no more numbers after the 3,5. No dots to show that it continues. So for multiple choice A, we can see that the 3 is repeating so it is a rational number. This decimal is actually the same 1/3. You can check it out on your calculator. 1 divided by 3. For choice B also we can see that we have two-seven, two-seven, two-seven. The two-seven is repeating so it is a rational number. Same here. This has one-two-three, one-two-three, and one-two-three. The one-two-three is repeating so it’s a rational number. For the last one. When we look at the decimals, we have 4142356 and so on. The numbers are not repeating in any orderly form, so we say it’s NON-Repeating. Also the number continues, so we say it’s NON-TERMINATING. We can therefore say that the 1.41421356 and so on is a irrational number.

This is actually the same as √2 which we already know from the previous question to be irrational. Try it on your calculator. Square root of 2. So this is our answer.

Please note that the calculator will give you to a certain number of decimal places. So some calculators will show 0.333, others will show 0.333333 and others will show something different based on the number of decimal places. The dots are not shown. Please Note also that usually unless you are being tested on your ability to know rational and irrational numbers, most GED questions write all irrational numbers in an approximated form, making it look like a rational number. Example, although π is an irrational number, 3.141592653 and so on, on the GED you are told to use 3.14. Making it look like a rational number, although it actually irrational. Let’s end this video here. Please share this video to help a friend and most important subscribe to our Channel and watch all the videos in this course. We will cover everything. You’ll become a master at this and Easily Pass your GED and say a permanent goodbye to your math problems.

Have a great day see you in the next video.