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### Operations on Numbers

- To add numbers that are the same sign, add the absolute value of each number together. The answer will have the same sign as the sign of the two numbers.
- Adding numbers that are different in sign can be done using subtraction. To subtract numbers, find the difference of the absolute value of the two numbers. The answer will have the same sign as the number that was larger in absolute value.
- When multiplying and dividing, if the numbers have the same sign the result will be positive. If the numbers have a different sign, the result will be negative.
- When raising a number to a power of a natural number, we can think of it as repeated multiplication
- When taking the square root of a number, we want the positive number that when multiplied by itself would give use the number under the square root.
- Square roots can be split up over multiplication and division. Be careful not to split up a square root over addition or subtraction.

### Operations on Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:06
- Operations on Numbers 0:25
- Addition
- Subtraction
- Multiplication & Division
- Exponents
- Bases
- Square Roots
- Principle Square Roots
- Perfect Squares
- Simplifying and Combining Roots
- Example 1 8:16
- Example 2 12:30
- Example 3 14:02
- Example 4 16:27

### Algebra 1 Online Course

### Transcription: Operations on Numbers

*Welcome back to www.educator.com.*0000

*In this lesson we are going to take a look at operations on numbers.*0002

*When you hear me use that word operations, I'm talking about ways that we can combine together numbers or do something to a number.*0007

*It gets very familiar things such as adding, subtracting, multiplying and dividing.*0015

*As well as exponents and square roots.*0020

*To combine numbers together, we use a lot of familiar operations in order to do so.*0028

*Again, adding, subtracting, stuff like that.*0033

*Note that depending on the types of numbers being used, certain rules applies that will help us put them together.*0036

*The rules that I will focus on are the ones that involve positive and negative numbers.*0042

*How should we deal with those negative signs? Let us see what we can do with addition.*0048

*When you add numbers that have the same sign then you are looking at adding their absolute values together.*0055

*The results if they do have the same sign or have the same sign as the original numbers.*0062

*In other words, if you are adding the other two positive numbers its result will be positive.*0068

*For adding together two negative numbers, then your result will be negative.*0073

*Now if you happen to have two numbers that you are adding together and they are different in sign,*0079

*then we will actually handle these using subtraction.*0084

*Wait for the rules on what to do with positive and negative on my next slide.*0087

*When dealing with subtraction and what we want to do is subtract the number that is smaller in absolute value*0095

*from the number that is larger in absolute value and looking at each of their absolute values taking the smaller one from the larger one.*0102

*When looking at the final result, the answer will have the same sign as whichever number was larger in absolute value.*0111

*If your larger number was or the larger number in absolute value is negative, then the result will be negative.*0119

*Remember that we will also think of adding negative numbers using subtraction in this way.*0128

*In different way we are writing it but instead we are really using subtraction.*0134

*The other two familiar operations that we have are multiplication and division.*0142

*The way we handle a lot of signs with these ones are by thinking of this.*0147

*If we multiply a negative × a negative then the result will be positive.*0153

*A negative times a negative results positive.*0158

*If we take two things that are different in sign such as negative times a positive, then the result will be negative.*0164

*Now these are the same two rules that we end up using for division.*0175

*If we divide a negative by a negative, you will get a positive.*0180

*Negative divided by negative result positive.*0184

*If they are different in sign such as negative divided by positive, then the result is negative.*0189

*Keep these in mind when working with that your signs, multiplication and division.*0198

*Let us talk about exponents.*0207

*When we have an exponent, we can think of it as taking a number and multiplying it by itself in a certain number of times.*0210

*For example, maybe I have 3 ^{7}, 7 would be our exponent.*0218

*I can interpret that as multiplying that number by itself 7 times or some other key vocabulary you want to pick up on.*0224

*The exponent is that number raised next of the 3, exponent.*0236

*In the base of the number is the number that we are actually being repeatedly multiplied, base.*0244

*Once we see what number that is and how many times we need to multiply it, usually we can go ahead and simplify from there.*0252

*There are some common exponents that we usually give other names.*0258

*If I'm taking 5 and I'm raising it to the power of 2, 5 × 5, this is often said 5 ^{2}.*0262

*Another good common one would be something like 2 ^{3}, 2 × 2 × 2 that would be 8.*0278

*I could also say that this is 2 ^{3}.*0290

*Key on these special words for some of these other powers.*0294

*Another operation that we can do with numbers is taking the root of a number.*0303

*The principal square root of a number is the non negative number (n)*0309

*that you know if I were to say multiply it by itself, I would end up giving that (n).*0313

*This seems a little funny special worry but let us see if I can describe it using (n) as an example.*0319

*Let us say I wanted to find out the square root of 25.*0326

*What I'm looking for is what number would multiply it by itself in order to get a 25? That has to be 5.*0330

*It is what I'm talking about here, positive number such that when it is multiplied by itself you get that number underneath the root.*0340

*For this reason we have a little bit of a problem with our negatives underneath the root.*0353

*After all, what number would you multiply it by itself in order to get -16?*0360

*We learned from our rules of positive and negative numbers that -4 and +4 would work to get 16*0364

*but unfortunately they are different in sign and we need them to be exactly the same, that is not going to work.*0373

*We are dealing with the principal square root because we are only interested in the positive numbers*0382

*that when multiplied by themselves would give us that number.*0387

*A perfect square is a number that is the square of a whole number and this one is usually reduced very nicely.*0393

*For example the square root of 9 would be the example of a perfect square, as it reduces down to a nice whole number, 3.*0401

*If you have a (a, b) being non negative real numbers,*0416

*then there are a few different ways that you can say combine or rip apart those roots.*0421

*3 multiplied underneath the same root and you can apply the root to each of those pieces.*0426

*If you are dividing and you have a root then you can put it over each of its pieces in the numerator and in the denominator.*0432

*You can use these rules in two different ways to simplify or combine words together.*0442

*We will see that a lot in some future lessons.*0446

*I'm pointing this out now so you do not make a common mistake.*0449

*Do not try and split up your roots over addition or subtraction.*0453

*We do not have a rule to do that yet or a good way to handle it.*0458

*In fact, in this example I have written below you can see that the two are not equal by simply evaluating each side.*0461

*9 + 16 would be a 25 and the square root of 25 is 5.*0469

*Looking at the right side square root of 9 is 3, square root of 16 is 4.*0475

*I'm putting those together I will get 7 and you can see that these things are not the same.*0482

*Be very careful when working with your roots.*0492

*Now that we know a few things about combining these, let us go through some examples and just practice with them.*0498

*The first one I want to add together a -3 and a -6.*0504

*I'm adding together two numbers that have exactly the same sign.*0510

*I will look at their absolute value and add those together.*0514

*The absolute value of 3 is 3, the absolute value of -6 is 6, if I combine those together I will get 9.*0519

*Since I'm adding together numbers that have exactly the same sign, the result will also have the same sign.*0531

*Adding two negative numbers my result is negative, -3+ -6 is a -9.*0537

*Moving on, 19 + a -12, I want to think of this as a subtraction problem since their signs are different.*0545

*How do I handle subtraction? Again I will look at their absolute value.*0558

*The absolute value of 19 and the absolute value of a-12, 19, 12.*0562

*I will subtract the smaller number from the larger number 19-12, what will that give me? I will get7.*0571

*I have to determine what sign this should be.*0584

*In subtraction we take the same sign as the larger number.*0588

*19 was larger in absolute value, it was positive, I know my result is 7.*0594

*If we do get a positive result as our answer, we do not write that positive sign up there.*0604

*We are just having 7, we will assume that that is a positive 7.*0609

*-8-11, an interesting way we can look at the problem, we could look at this as adding a -11.*0614

*The reason why I have looked at it in that way is that I could use my rules for addition.*0626

*If I'm adding things that have the same sign, I have looked at the absolute value of each of them, 8, 11.*0631

*Then I can simply add up those two values and get 19.*0640

*Since I'm adding two negative numbers, I know my result will also be negative, -19.*0645

*Let us try another one, 8 - -13.*0651

*When you subtract the negative this is another good situation that you could end up rewriting in a much simpler form.*0657

*When you subtract the negative, you can change it into addition.*0662

*This is 8 + 13 and now I'm adding together two positive numbers.*0667

*You made a positive 21 and one more, negative the absolute value of a -4 + 9.*0673

*Let us start inside those absolute values and see what we can do.*0683

*I'm adding together two things but they have different signs.*0687

*Let us look at the absolute value of a -4 and the absolute value of 9, 4, 9.*0692

*We want to subtract the smaller value from the larger value, 9-4 and that result would be 5.*0699

*I know what sign should that have, or the number that is larger in absolute value is 9.*0708

*And that was a positive value over here, I'm looking at a 5.*0716

*There are a lot of other things I have left out here so far,*0721

*those would be the absolute values and that leading negative sign.*0724

*Let us go ahead and put those in there now.*0727

*I want to do the absolute value of 5, all that is simply 5.*0731

*I still have that negative sign hanging out front and it is been there since the very beginning.*0740

*I can see that after done evaluating this one all the way, my answer is actually a 5.*0743

*Let us work on adding, multiplying and dividing the following numbers.*0752

*We only have one rule to take care of that is when we are multiplying together two negative numbers, we get a positive.*0757

*If we are multiplying together two numbers that are different in sign, the result should be negative.*0766

*4 × -7, I just want to think of 4 × 7 that would be 28.*0775

*Since they are different in sign, I know that this will be a -28.*0784

*Moving on, -6 × -5, 6 × 5 would give me 30 and now here I have a - × - I know the result will be positive.*0789

*But again we usually do not write that positive sign in there so just leave this as 30.*0801

*-12 ÷ 3, 12 ÷ 3 would be a 4, negative ÷ positive would be negative, our result is a -4.*0806

*One last one, -60 ÷ a -5, 60 ÷ 5 goes in there 12 times and negative ÷ negative is positive, my result is a 12.*0820

*You need to be very careful with these rules for multiplication, make sure you have these memorized.*0835

*Let us do a few involving our exponents.*0843

*Remember we can think of these as repeated multiplication.*0846

*2/3 ^{4}, it can be the same as 2/3 × 2/3 × 2/3 × 2/3, we are doing it 4 times.*0850

*I simply multiply it across the top and across the bottom 2 × 2 × 2 × 2 would be 16.*0862

*Then I have 3 × 3 × 3 × 3 = 81.*0871

*All the numbers here are positive so I know my result is positive.*0876

*Here is a tricky one, -5 ×-5 × -5, let us take this two other times so we can what is going on here.*0881

*Here I have a -5 × -5 the result of taking 5 × 5 would be 25.*0896

*Taking a negative × negative I would know that this would be 25.*0904

*That looks good, let us go ahead and work in this last value of -5.*0910

*We want to multiply 25 × -5, the result there would be 125.*0918

*Since I'm multiplying a positive × a negative result is -125, -125 would be my answer.*0929

*The next one looks very similar but it is actually very different.*0939

*That one is 2 ^{3} and that negative sign is just out front of all that.*0943

*We recognize that the 2 is the base and that the negative is not included in that base*0949

*since there is no parenthesis given to group it in that way.*0956

*We want to figure this one as 2 × 2 × 2, I have multiplied it out three times.*0960

*And as for what to do with that negative sign, if we have not put it up front it is a long pretty ride.*0966

*2 × 2 × 2 that would give me an 8, all of those numbers are positive, 8.*0973

*Of course let us put our negative sign out front since it was out front at the very beginning.*0979

*We can see that our result is -8.*0984

*Now on to some square roots, these ones we simply just want to break them down and simplify them as much as possible.*0990

*The first one I have the square root of 64.*0998

*Think for yourself what number would you have to multiply it by itself in order to get 64?*1001

*I have a couple of options that could be 8 and 8, that would have given me 64.*1008

*Or it could be -8 × -8, that would also give me 64.*1013

*We are only interested in the positive values that do so, let us not worry about those -8.*1018

*We will say that our answer for the square of 64 is 8.*1024

*Moving on, the square root of 169 ÷ 81.*1031

*This one we can use one of our rules and break up the root over the top and over the bottom.*1037

*Now we can end up taking the root of each of these individually.*1045

*What number multiplied by itself would give us a 169? That have to be a 13.*1048

*A number that would be multiply by itself to get an 81? 9.*1055

*The answer in this one is 13/9.*1060

*Continuing on, this one has a negative sign out front but that is not underneath the root.*1064

*I would not worry about it just yet instead let us just focus on the square root of 36.*1071

*That would be 6, of course we will go ahead and put our negative sign and see that our final result is a -6.*1079

*One more, this last one involves the square root.*1091

*What two numbers when multiplied together would give us a -49 remember they must be the same.*1095

*We got a few problems, do not we? If I try and use 7 and 7 that would give me 49, that does not work.*1102

*If I try and use a couple of-7, that does not work, that still gives me 49.*1109

*I can not use one positive and one negative even though those give me a -49.*1114

*Those are not the same sign, one is positive and one is negative.*1121

*What is going on here, if you remember about your types of numbers, these are imaginary numbers.*1125

*I will leave that one just as it is until we learn about simplifying imaginary numbers in some later lessons.*1137

*Some various different ways that you can go ahead and combine numbers using some very familiar operations.*1144

*Remember that most of the rules that I covered will give you some tips on what to do when they are different in signs.*1150

*Positive, negative, negative, positive and all those will be handy in figuring out the overall sign of your answer.*1155

*Thank you for watching www.educator.com.*1166

1 answer

Last reply by: Professor Eric Smith

Sun Dec 3, 2017 5:52 PM

Post by Genevieve Carisse on October 19, 2017

You're an amazing Prof! Thank you for your lectures!

1 answer

Last reply by: Professor Eric Smith

Wed Mar 18, 2015 10:28 PM

Post by antonio cooper on March 18, 2015

I was wondering since I see some other math prof. create quizzes I was wondering if you were planning on doing this as well? Thank you very much for what you do.

1 answer

Last reply by: Professor Eric Smith

Mon Jun 16, 2014 3:21 PM

Post by C. Barnes on June 13, 2014

WHOA!! in example number 1 where you have a positive 8 -(-13) wouldn't that result in a -5 instead of a positive 21? I believe that the sign follows the larger number when there is a positive and negative combination. Ie, 2 -(-3)= - 1. If I am incorrect, please clarify>

1 answer

Last reply by: Professor Eric Smith

Thu Oct 10, 2013 2:02 PM

Post by Ezuma Ngwu on October 4, 2013

Is there any easy way to calculate exponents without a calculator?

2 answers

Last reply by: james templeton

Mon Apr 27, 2015 10:25 AM

Post by Abhijith Nair on August 20, 2013

Does this course cover everything found in an Algebra 1 course?