GRE Arithmetic – Exponents
All right for today we`re going to talk about exponents exponents is a gateway to higher-level math that we`ll see in other chapters this will be the ending of the arithmetic chapter so now let`s talk about exponents I`ll dive right in and show you how we write exponents so let`s look at 3 to the 2nd power yes I said 3 to the 2nd power this 3 is what`s called the base and that 2 is what`s called the power so if I have 3 and that`s 2 above it that means that we have 3 to the 2nd power or in other words another way of saying that 2 is not 2nd power but squared.
Whenever you have a number to the second power we can say it`s squared so this would actually read 3 squared this is 4 squared this is v naught squared but cubed you see this power can change to any number so now this would be cubed this will be 5 to the third power or 5 cubed great now let`s find out what this even means well regular numbers or numbers with powers that you don`t see look like this 3 5 6 we`re familiar with these integers but actually they do have a power their power is 1 since it`s 1 we don`t care and we don`t put them there but they still are there even though we don`t show it all this means is when you have a number a and a is our variable for any number and you take it to some power I`ll call that power N and we`ll get familiar with working these variables but if you have a number a or an integer a any number to the power of n that means that we`re multiplying a by itself and times ok we`re continuously multiplying a by itself and times so if we have something like a cubed a cubed then we`re going to have that equal 8 a that`s two times times another a and so that`s one two three times great and now looking up at what I put before what I talked about before this three squared 3 squared would be three times three equals to nine so yes three squared is nine and that`s the basics of exponents once you understand this you can do many more problems such as let`s let`s take a look at this op this expression.
Let`s have six squared minus six plus six cubed okay so we have six squared six squared is 6 times 6 or 36 minus 6 well 6 to the first is just 6 plus what 6 cubed well let`s take a an aside here and find out what 6 cubed is so 6 times 6 times 6 6 times 6 is 36 times another 6 what is 36 minus a times 6 so that would be 216 great now we have plus 12 216.
Now we just have to do 36 minus 6 plus plus 216 so 36 minus 6 is 30 plus 2 16 is 246 so that`s our answer this number here is 246 great so that was an expression with exponents and now we can have different processes like cubing and squaring and those are just words that represent taking a number to the second power or taking a number to the third power all right awesome and so I`ll just write those words down so we`re familiar with the the terminology so if we wanted to write three squared with sweet to the second power so this is three squared and this is what it`s called squaring 3 we`re taking the square of 3 so we`re squaring 3 so I`ll just write so that`s squaring 3 ok great and QB 3 would be similar 3 cubed and now what if we wanted to go backwards we know that 3 squared equals 9 so I`m gonna write a 9 here but what if we wanted to do something called taking the square root square I`ll write this root in blue taking the square root of 9 so taking the square root of 9 is doing the opposite of taking the square root of 9 if 3 squared is 9 then the square root of 9 would be 3.
The square root of 9 of 9 is is 3 of 9 is 3 okay and we know 9 is 3 squared so we can take that say the square root of 3 squared is 3 now we`re talking about math we can`t write out English words all the time so how do you represent taking the square root of a number we represent the square root of a number of a number by this so the square root this means the square root.
This is a sign used by the scrub to represent that and now for the number we put that number inside the square roots we put the number in there so if we wanted the square root of 9 that would be radical 9 alright so yes this square roots can also be called radical so I`ll write that here radical or square root so 2 acceptable words or phrases for this would be the square root of 9 or radical 9 and now we know that equals 3 we know that equals 3 because 3 times 3 equals 9.
For a square root you`re finding what number multiply is multiplied by itself equals this number inside the square root so the square root of 16 would be something times something equals 16 well that number is 4 because 4 times 4 is equal to 16 all right and now we can try even the square root of 36 with equals 6 because 6 times 6 is 36 all right then we can talk about something that`s not a square root but a level up which is the cube root all right and as the name sounds it`s the cube root meaning taking the third root of a number and so we don`t represent it by this square root but we put a 3 here to represent cube root ok so what do we get when we have the cube root or try and find the cube root of 27 so it`s not what number multiplied by itself equals 27 its what number multiplied by itself three times equals 27.
What number times itself 3 times equals 27 that`s 3 because 3 times 3 times 3 equals 27 all right great and you can keep on going you can take the 4th root of a number the 5th roots and so on alright great and that`s how you take the square root of a number so now I`m going to go over how to take powers of negative numbers all right so before that this is what we talked about we talked about taking the power of a where a is just a number we talked about taking the roots of a to the N and then we talked about taking the third root of a to the N and so on.
We talked about this so far and now let`s talk about how we take the same operations but for a negative number so if we have or the opposites of these operations for negative numbers so if we have B to the N where B is less than 0 negative numbers are less than 0 then we get we get a couple of things it depends on B ok so what if we have negative 2 squared that would be 2 negative 2 times negative 2 which is 4 that`s pretty obvious what if we have negative 3 cubed well that will not be 3 times 3 times 3 which is 27 that would be negative 3 times negative 3 times negative 3 which would equal negative 27 so the rule is I`ll write this here if n is even then a or B to the nth power even if B is negative will be positive so for positive a write a plus sign and that works for fractional numbers too so if n is even then a B to the N is positive no matter what sign B is or what number B is and great so now what if n is odd you guessed it then B to the N would be negative.
Great so that means that negative 4 to the third power is a negative number 4 to the third power is a positive number all right and whenever you`re taking numbers to some power you just multiply it by itself that number of times if it`s an odd number of times it`ll always be negative great so we went over a couple of other rules or some tricks I mean you could have figured this out just by doing the work but it`s great to know that you should be getting a certain sign once you`re doing these problems all right and now what`s another way of writing negative 3 to the third power a negative number to the third power so you just write it as negative 3 times a negative 3 times negative 3 but the interesting part is when you take a number to a negative power all right so we know to take negative numbers to positive powers what if you have B to the negative power so n is positive but negative n is negative of course that you would just do as take the coefficient here the coefficient here is a 1 it`s an implicit one we take that coefficient write it on the numerator and then on the denominator we write B to the positive and power.
We ignore this negative and these guys are the same thing okay so let`s try an example what if we have 3 to the negative 3 power well we know 3 to the third power was before was 27 and 3 to the negative third power or negative 3 to the third power would be negative 27 what about 3 to the negative third power or to the negative 3 so that would be 1 over 3 cubed and so that`s 1 over 27 awesome so that equals 1 over 27 you see it`s a reciprocal now what if we have something like how about negative 4 to the negative 4 power well that would be negative 1 since this coefficient is negative 1 divided by or to the fourth power while we were fine four to the fourth power and that would be four times four times four times four and that would be 256 so we would get negative one over 256 which is some fraction or some decimal great now that we know how to do that let`s solidify that rule.
This is the rule B to the negative and power is one over B to the N the general rule is I`m going to write an as a coefficient of a number and X is a variable so in this case in this case X would be 3 and a would be this one here.
A is a coefficient just so you guys don`t forget this is what the coefficient is and X is the variable and n is, of course, the power which we`re making negative so this equals an over X to the N power wonderful this is the general rule that you should either copy down or just or just memorize alright and as we do more problems it`ll become clearer so I`ll give you a couple of other rules that should be obvious but it`s nice to have them down.
These are some rules let`s look at number one and some of these will be the ones that we`ve done so radical a or square root of a squared would be a why because we know that these square roots of a number is going to be the root of the number or the number that when multiplied by itself by itself gives you a so remember before we have the square root of 9 that is 3 because 3 times 3 gives you 9 but what we`re doing here is taking the square root of 9 are all right 9 here 9 and then we`re squaring it.
Then what does that mean that means that the square root of 9 is 3 and if we square it that`ll just give us back 9 so we can write this rule that the square root of 9 squared equals a and in this case a is 9 so the rule is the square root of a squared is.
The square root of B squared is B whatever number a is you take the square root of it and then square it and you know that these two things cancel out those two things that are the square roots and a square.
Those cancel out when you`re taking a square of the square root and now what if we have the same thing but for cubes so if we take the cube root of a to the third power then that would just be a again all right so this three has to match this three and they cancel out and you just get a so the actual general rule here which I`ll box is well we need to make a general number a general variable for three so that would be the route would be beep.
Then we have this root sign of our number inside to the B power would equal an all right so that`s our rule great so now rule number two and you can practice this and see if it works out so rule number two would be that square root of a squared also equals a and that`s the same thing you can tell all right and I`ll just I`ll just show you so before we had here we had that the square root of 9 is 3 but now what if we had the square root of 3 squared .
These square roots of 3 squared would be here so the square roots of 3 squared well 3 squared is 9 so the square root of 9 is 3 or you could just do this you can just imagine that this 2 and the square root sign cancel out and of course that`ll work for cubes and so on so the general rule now is that the or that the cube root of a 2 well instead of me writing cubed let me write it as a general rule which I said I would do so we have the nth root of a to the N power equals a and really doesn`t matter which variables you use they`re just going to be filled in with numbers.
The nth root of a to the N equals 8 awesome so now on to our third rule number three we have radical a times radical B so what happens if you take the square roots of two numbers and multiply them and then we get the square root of a B so yes an example of that would be the square root of nine times the square root of sixteen that would equal that would equal three so the square root of nine times 16 and 9 times 16 would equal what is that equal well we know whatever it equals it has to be equal to a B so 9 times 16 is equal to 144 and the square root of 144 is 12 all right and to double-check that we have that the square root of 9 is 3 the square root of 16 is 4 and 3 times 4 equals 12 so yes 12 equals square root of 144 which also equals 12 all right great so that`s another rule and now let`s talk about a fourth rule we have that what if we have the square root of a fraction so the square root of a over B equals the square root of how many right this better so the square root of a over B equals the square root of a over the square root of B so like the rule says we have let`s try the square root of 4 over 9 which equals the square root of 4 divided by the square root of 9 which equals 2 over 3 so the square root of 4/9 is 2/3 and of course we can generalize this to any nth root for example what if we take the cube root of 27 over 64 that would be the cube root of 27 over the cube root of 64 and now that would be 3 over 4 so the cube root of 27 over 64 is 3 over 4 great now we can talk about other ways to do these problems I`ll just show you guys one other one other trick ok so before we had the square root of a-squared well whenever we have the square root of a number that`s the same thing as saying that number which is a squared to the one-half power.
This square root means to the one-half power and whenever you have something like this the rule is that if you take a number to a power you multiply the power of this number times the power of this so two times one half is just one so this is the same thing as a to the first power which is a and that`s why that`s why the radical a squared equals.
What we`ll go over this in more detail later on but you have the basics down now which is great so we`ll move on to a challenge question next all right so here are some problems for the exponents part the exponents chapter that we did before so the first one what is radical a times radical B times four times radical eight.
Radical a times radical B times radical four times radical eight give me that answer number two what is radical 81 times the cube root of 27 to the ninth power over radical 120 so give me that answer so you guys should pause the video try it out and see if you can get the answers and then come back alright here we are back again let`s go over number one I`ll do this in red.
Here`s number one so we have radical a times radical B times the square root of four times the square root of eight and what is that equal okay so remember the rule that I`ll write that here in blue this is just a rule radical a times radical B is the same thing as radical a B and this is where it comes in handy because now we can combine these numbers because it`s not always the case that you get the square root of a number sometimes you need to combine them and in this case we want to simplify so we need to combine all of these guys so radical a times radical B times radical four times radical eight we multiply everything inside to get this so we have 4 times 8 is 32 so 32 a B and that is our answer but but we can simplify it more it`s ok if you don`t in this case because we`re just starting out but I`m going to show you guys what the proper simplified answer is so I want to take these apart even though I even though I condensed them the reason I want to take this apart is because I want to have a part that is a perfect square meaning that if I take the square root I`ll get an integer and a part that isn`t a perfect square.
So as on the side as in the side I`ll get back to this a perfect square our numbers like 16 81 9 why because the square root of 16 is 4 this and that`s my operation the square root of 81 is 9 so let me just.
Then the square root of 9 would be 3 all right so you see that if you take the square root of these perfect squares then then you get an integer but one thing one number that`s not a perfect square would be something like 32 you can`t take the square root of 32 and expect to get some number that`s an integer so what you have to do is break it apart well if radical a times radical B is radical a B then it`s fair to say that radical a B is radical a times radical B we can go backwards in that case what I want to do here is then take this square root of 32 and rewrite it as two numbers that multiply to 32 rewrite it as a product of its factors well we know that 4 times 8 is is 32 but we can go even better we know that 16 times 2 equals 32 and so radical 32 would actually be radical 16 times radical 2 and we know radical 16 is 4 so this would be 4 radical 2 all right radical 2 is not a perfect square so we keep it as that and so why did I show you guys this I showed you that because now we want to break this apart and we want to write this as radical 16 times radical two times radical a B because this becomes tene it becomes 32 and so we can rewrite that as four radical two times radical a B and of course we don`t want these separate radical signs we just want to combine everything so this would be four radical two a B do you see how we combined this square root of a B and the square root of two we combined what was inside and got square roots of two a B and we kept the fort outside all right there are multiple ways of doing this for example you could have just changed this square root of four into a two but I wanted to show you how you can condense and then expand all right so it`s a it`s a process of condensing and expanding until you get the condensed form that is the simplest all right so when they say simplify you`re going to get the most condensed form as possible you do not want to have all of these different terms you want to have one term in that term that we got here is four square roots to a B all right great and so that was just the first one now let`s go over number two I`ll copy down number two here so let`s see we have a radical eighty-one so we have number two as square root of 81 times the cube root of 21 sorry about that the cube root of 20 27 to the ninth power I believe and so yes and then all over square root of 120 all right let`s do the numerator first we have the square root of 81 we know will be 9 the cube root of 27 is going to be 3 and 3 to the ninth power is some number but we can do better than that we know that square root the cube root of 27 is actually 27 to the one-third which I explained in the previous video you just take whatever roots it is and reciprocate it just like the square root of 4 is 4 to the 1/2 so the cube root of 27 is 27 to the one-third but don`t forget that 9 that we have on top and now we divide it by the square root of 120 ahright great so now let`s even let`s move further 9 this guy this guy here what is 27 to the third time to the ninth power well these feet two things multiplied so 1/3 times 9 is actually 3 so we have 27 cubed awesome and then over square root of 120 and as in the side you should always simplify every term before you start multiplying things out so what I mean is I don`t want to start trying to find out what 27 cubed is before I simplify square root of 120 because you may get some things that cancel out and you don`t want to miss that opportunity because it saves you work all right so we we copy the top so that becomes 9 times 27 cubed and now this square root of 120 let me just bring this to the side this square root of 120 is something that I want to break apart to expand into maybe a perfect square if we can find one so 120 what is there a perfect square factor of 120 yes there is isn`t 120 the same as 10 times 12.
Once when is the same as 10 times 12 and these two are the same and then we can break this apart as radical 10 times radical 12 all right great and now we don`t have a perfect square yet so we can go even further we can say that we can say that the square root of 12 is the same as the square root of 4 times the square root of 3 because isn`t 4 times 3 12 yes it is and bring down this square root of 10 so now we have the square root of 10 times the square root of 4 times the square root of 3 or any of these perfect squares this is so we have 2 times the square root of 3 times the square root of 10 all right and we always simplify by putting the integer first so we have 2 square root of 30 all right that radical 3 times radical 10 becomes a radical 30 and there`s no factor of 30 that is a square root only factors of 30 we have our 2 5 6 15 and 30 none of those are square roots and 1 but we don`t we don`t include 1 all right so now we have on the – square root of 30 awesome so now we can really simplify all right so unfortunately this doesn`t not work out so beautifully but we can still write what the answer is we have nine halves 9 over to 27 to the third power what is that so we have 27 times 27 times 27 and that`s 19 thousand six hundred and eighty-three divided by two times radical 30 and sometimes it won`t be beautiful.
We just find out what nine times this number is and that would be 177 thousand one hundred and forty seven divided by two radical three or two radical 30 awesome so that is the answer we can go even further just by dividing that number by two if you want and so we get this is eighty-eight thousand five hundred and seventy three point five and so we have eight eighty five or 88 thousand five hundred seventy three point five divided by the square root of 30 all right awesome and so that`s the