all right welcome to my youtube channel viewers um if today is your first time of coming to my videos please consider subscribing hit the subscribe button the red button just below this video and also use the thumbs up that's the like button thank you very much so in today's class we are going to be looking at indices as a topic okay this is a very beautiful concept of mathematics which is actually an english word so what is indices synthesis is just the plural of the word index the plural of the word index which actually means power and you can also call it exponent so the mathematics of indices is where we study numbers and they are powers numbers and their exponents and what does that mean so for example if i have say 2 raised to power 3 now so this number now is raised to a power of 3 and so this is called the base why this is called the power and like i said this is the power or the exponent and you can also call it the index of two and what does that mean two raised to power three actually means that is two multiplying itself up to three times that's 2 times 2 times 2 the number of power you have and so if i say a raised to power 3 let's say 2 for example that is a times a 2 times so if i then say a raise to power n so this means a times a times a times a up to n times so that's how we write it so you are multiplying it by itself up to n times that is the number you have as the power and so that's what we mean by uh indices the study of numbers and their index the powers okay and now for us to effectively do these indices our study numbers and their powers would have to learn how to you know do operations basic operations of numbers with powers and for us to be able to do that we need what we call the rules of indices we need to know the basic pattern of doing these operations okay and so we are going to look at the rules of indices what are the basic rules of indices they include the following now we have what we call the multiplication rule that is if i have a raised to the power n multiplied by a m multiplied by a raised to the power n what is that going to give me please quickly you will notice that if i have for example a raise to power 3 times a raised to power 2. what does this thing mean it actually means a raise to power 3 is the same thing as a times a times a and then times while a raised to power 2 is a times a that's two times so you see that if you remove this bracket you will simply get a raise sorry a times a times a times a times a and so you're going to get a multiplying itself how many times five times which is actually a raise to power five so you will simply see that a raised to power 3 times a raise to power 2 is just the same thing as a raise to the power 3 plus what 2 which is the same thing as the areas to power 5 that we have here and that gave us a general rule for multiplication of numbers with powers when the bases are the same so if the bases are the same you simply pick one of the bases and then just add the powers and so that's what we call the multiplication rule so it obeys and then what about the second rule which we call the division rule so the same thing is applicable here if i use this same example here you would see that this is areas part three all over a raise to power 2 this is simply going to give me a times a times a times a into 3 places and then this one will be a times a now by division a will take away a and a will take away and here i only have just a which is the same thing as a raise to power one right and so you would see that division is just a normal thing as doing your subtraction so instead of doing this division i would simply have done a raise to power three minus two which will give me a raise to power one and so that gives us the general rule of division as a raise to power m minus n okay and then the third rule we have what we call the power power rule and so if i have a raised to the power m all raised to power n this is the same thing as a raised to the power m times n and so you can as well see an example for that now in the fourth one we have a raise to power zero whenever a number is raised to the power of zero this is the same thing as one and of course you can see that here for instance if i say a raised to power zero you will see that this guy is actually the same thing as a raise to power say let me use a number now say three minus three and that is true and of course you can see that three minus three is zero so this is the same thing as a raised to the power zero and so but i know that by this rule the second rule here a raised to the power three minus three is the same thing as a raised to the power 3 all over a raise to power 3 and that is going to give me air is a times a times a times a a times a times a times a and this will take away this this will take away this and you eventually get one so any number raised to power of zero is the same thing as what one what about the fifth rule when i have a negative index what happens to that you would see that this is the same thing as one all over the positive index so the negative power here will change to the reciprocal and then you will return areas to the power of the positive end instead of negative and that you can as well see that of course a raise to power let me use our negative n as an example this thing is the same thing as a raised to the power 0 minus n if i do this subtraction i'll get back my errors power minus n number by this second rule also you will see that this is the same thing as a raised to the power 0 all over a raised to power n and area to power zero by the fourth rule here is the same thing as one and that would be all over a raise to power and which is exactly what we have here and then the fifth rule or sorry the sixth rule that we're going to look at here is um the rule of um a fractional power what if i have uh let's say a raised to the power 1 over n this is the same thing as a to the nth root and in general in general if i have a raised to power m over n this is the same thing as the nth root of a or raised to the power of m so what happens here is that your numerator will turn to a root and sorry your denominator will turn to a root and your numerator will turn to a power so in this case here in this special case where the numerator is one of course if you take this to the power of one you will get back the same and that's why it is this way that is why whenever you have for example let's say um a raised to the power half this is always square root of a because square roots of course you can write that 2 is always invisible there if you have air is per 1 over 3 this is the same thing as the cube root of a okay so now we call this um the fractional index rule if you have the power as a fraction the numerator of the fraction will be a power the denominator will go as a root and um okay that's the sixth truth sorry the sixth rule that we are going to see remember these rules are not in any other there is no other okay so the seventh one that we are going to see here says that if i have the multiplication of a and b raised to the power of m that this is the same thing as m sorry a raised to the power m times b raised to the power m that means the power is distributive or multiplication it can distribute on anything multiplying themselves and this can actually be in two places i can say the second part is division it is also distributive on division that means this is going to give me a raise to the power m all over b raised to the power m now but this is not true for subtraction and addition and so remark that note this is not true for i i saw uh addition and subtraction that means if i have a plus b or a square m this is not the same as a raised to the power m plus b raised to the power m always note that and it also holds for subtraction it is power it's not distributive on additional subtraction that means this is not the same as a power m minus b power m please take note of that but that holds for multiplication and for division okay and then the last rule we are going to see here um is that if i have a raised to power m to be equal to um a raised to power n that means two numbers two numbers in index form having the same basis that means the base is the same and they are equal that if this happens then their powers must also be what equal please take note of this and so these are the basic rules of indices that we have so we have the multiplication rule division rule the power power rule we have the zeroth rule sorry the zero power rule we have the negative index rule the fractional index rule we have the um when you have the power of products power of quotient and and then we have when you have them two index numbers with the same base being equal their powers will also be equal now quickly we are going to take a few examples but before we go to examples please take note of this when you have a negative number note [Music] when you have a negative number raised to a power of say m now this particular number now is the same thing as positive a raised to power m if my m is even and it will give me the negative a raised to the power m if my m is odd an example of that is when i have for example one raised to power say two this is simply the same thing as one raised to power two so because this will give you one and but if i have minus one raised to the power three that is telling us that this is minus one raised to the power three and one raised to the power three is one which will give us minus one and that was for any number at all if you have negative 3 say raised to power of let's say 4 this is the same thing as 3 raised to the power 4 because 4 is even and this is going to give us what 81 and of course we know that if you have now negative three say raise the power three three being an odd number this is going to give us minus three raised to the power three and three raised to the power three is twenty seven so you get minus twenty seven and so this particular of course it should be a rule it can always guide us when we are raising a negative number to a power so if the power is even just remove the negative sign and take the power you get the same answer but if the power is odd you just remove also the negative sign take the power and then attach the negative after you've gotten an answer and that is the solution now we are going to take just few examples on how to make use of these rules to solve problems how do you use these rules to solve mathematical problems let's look at few examples so problems okay here we have number one problem we have here says evaluate the following the example says evaluate the following and we have two x's per half multiplied by 2x power 3 all raised to the power of 3 over 2 so this is going to give me 2 x to the power of half if i open this bracket i am going to get this power will distribute you remember that power distributes on multiplication so it will distribute here so i'll have 2 raised to power 3 over 2 multiplied by x raised to the power 3 times 3 over 2 because that one already has a power and so that is going to give us and us in fact we are supposed to do the same here so let's quickly do that if we do that here we are going to have two raised to power half then multiplied there's a multiplication here x raised to the power half also so because the power we distribute and when that happens this is going to give us two raised to the power half times x is the power half times 2 raised to the power 3 over 2 times x is the power this will give us 9 over 2 and of course we can you remember multiplication is commutative so two has the same base as this so let them come together let the ones that have x go together so this is going to give us so to multiply these two i would just pick one of the bases and then add their power so i will have one half plus three over two multiplying x i will do the same here i will pick one of the bases and then add up what i have here which is one half plus nine over two and that's going to give me if you do the addition here you are going to have 2 raised to power 4 over 2 and then you are multiplying by x raised to power 10 over 2 since they have common bases sorry command denominators you just pick one of it and then add the numerators and this is going to give us 2 raised to the power 2 times x is the power 5 and so our answer is 4 x raised to power half and that's what you have as a solution to this first problem that we have here the second one that we want to look at that we are asked to evaluate is um look at it here we have six to five raised to the power of 1.5 so what do you do you can see here now the power is decimal whenever you have a decimal power now the rules we have read now only handled fractional power so if i have a decimal power my own effort my my target would be can i change that power that is in decimal to be in fraction and if i can do that i can then apply the fractional rule or the fractional power rule so this is going to give me six to five my one point five is the same thing as three all over what two and by fractional index the denominator will turn to a root and then the numerator will become a power so this will simply give me square root of 625 which is what 25 r raised to the power of 3 and that is your answer so which is just the same thing as 25 cubed so when you take the cube of 25 you will get the answer to a problem and so that is the second issue now the third one we have a third example which is c sorry we have 0.04 just to put it in another form and if we do that we also have a mixed number and there is no recovering for mixed number like i said earlier there is a rule that cover for a fraction and so what do i do i'll change this into a fraction and that's and even this decimal here should be changed to a fraction and if i do that here i am going to get four all over 100 and that will be raised to the power of this is minus 3 over 2 and the negative index rule says that when i have a negative power that it is the same thing as 1 all over 4 over 100 r is the power of 3 over 2 and this is going to give me 1 all over remember the fractional power the denominator becomes a root and so i have the root of 4 over 100 rs to the power of 3 and this is going to give me 1 all over the root of 4 over 100 is 2 over 10 because the root of 4 is 2 the root of 800 is 10 and everything is raised to the power of 3 which is going to give me 1 all over 2 raised to the power of 3 remember that power will distribute on a fraction 2 raised to the power of 3 will give me 8 and 2 is 10 raised to the power 3 will give me 1 000 and when you take the reciprocal your final answer will be one thousand over eight and that will give you 125 as your answer okay so and that is the solution to the third problem that we have there okay now the next problem we are going to look at here says that we should okay so this problem says express the cube root of a b c raise power minus four all over the fourth root of a cube b response knows three c um with positive indices meaning we should simplify that expression we have there and let all the powers of every variable there be positive index and so let's try to do that so solution what do i do we apply the rule the first one is that cube root means a power one over three and so i'm going to get a b c raised to the power four or raised to the power of one over three all over now the denominator there will become raise to the power of one over four and that is a cube b minus three c raised to the power of one over four and then what is the next open up the bracket when you do that the power will distribute on all of the variables there and so i will have a raise to power one over three b raise to the power one over three and when it distributes on this you have minus four times one over three which is minus four over three so i'll have c raised to the power minus four all over three and then all over the same will happen here this will become three times one over four which is three over four and this will be four raised to the power minus three over four and this will be series to power one over four and so with that what should i do next all i need do next now is to now apply the division rule remember the division rule here i have a divided by these this divided by these this divided by this so i'm going to apply that now and if i do i'm going to have a raise to the power one over three minus three over four because there is a divide a division of a raised to the power three over four then times b raised to the power one over three minus also number the power of this one is negative then that is another minus three over four and then c times c raised to power minus four over three minus one over four and so we will try to simplify this and to do that one over three here you will have the lcm to be um 12 and if the lcm is 12 you will get 4 minus 4 minus 4 into 12 is 3 that's minus 9 and that's going to give you minus 6 over 12 then b will become remember that you will have one over three plus three over four lcm is also 12 and this goes here will give you four and um this one going here will give you three also three into this is a 9 and that will be plus 9 which is uh 13 over 12 so we have 13 over 12. sorry the the the first part is supposed to be sorry four minus nine which is five not three minus nine sorry it should be four minus nine over twelve okay so that's going to give us a minus five over twelve thank you okay so now what about this one we there we are going to have minus 4 over 3 minus 1 over 4 12 is also the lcm this is 4 minus 16 this is 3 and we have 3 there that's minus 19 over 12 so we have c raised to the power minus 19 over 12. now but the question is asking us to ensure that everything is expressed in positive index so what do i do i will apply the negative index rule here and also here and if i do that i am going to quickly get um b is the only one positive so it should be the only one remaining up then for the new denominator if this guy comes down is going to become 1 all over a raised to the power of 5 over 12. so it will come down as a denominator so i'll have a raise to power 12 5 over 12 and that will happen to c also that will become 19 over 12 the negative signs of the powers will disappear and so here what do we then have we are going to now get um so you will see that sorry remember that this is 13 over 12. so you will see that all of them now have common denominator as their powers sorry yes in the powers the fractional powers have common denominators that means i can bring out 1 over 12 from all of them if i look at what i am saying this is going to give us the same thing as b raised to the power 13 times 1 over 12 and the denominator will be a raise to the power of 5 times 1 over 12 and so i can factorize out that power of 1 over 12 and that's going to give me b raised to the power 18 all over a raised to the power 5 times serious power 19 all raised to the power of 1 over 12 and i can as well write that as the 12th root of b raised to the power 13 all over a raised to the power 5 times c 0 to the power 19 and that is my answer so i've been able to simplify the initial expression i am giving which had negative powers and now i have a single expression where all the powers are positive and that is the solution to that problem okay and that's the second problem which is actually the fourth so that in my numbers it is the second so here we take the third now the third problem here says without the use of tables okay so we here we have without the use of tables that's mathematical tables evaluate this and this so they are not expecting us to do this using the normal log tables or whatever to do the evaluation they are expecting you to use the rules of indices to solve this problem i am going to do a and then b is going to be your exercise and i expect you to put your answers in the comment section below okay so when you are done please drop your the comments of your solution to this problem ask your comments in the section below i would want to see what your solution will be so let's take the first problem here so here we have solution the a part that says that this we have the square root of 0.81 times 10 raised to the power minus 5 all over 2.25 times 10 raised to power 7 so what do we do with that so i would start by first of all trying to see that since i have a decimal number here can i make it become a whole number anna there's something beautiful about this if you look at the numerator there you have 0.81 and that's if i can take away that decimal number the number we just have there will be 81. and 81 is a perfect square okay so that means and if you look at the denominator you see something similar you have 2.25 and if i remove this decimal point i will get 2 to 5 and it's a perfect square root that means if you take the square root you give it you get a whole number and so what do i do then i will try to find a way to take away that decimal number and how do i do that if you go back to a topic we call standard form you will see that 0.81 is actually the same thing as 81 times 10 raised to the power minus 2 and how is that gotten we know that 0.81 is the same thing as 81 all over 100 and we know that 100 is 10 raised to power 2 and by a lot of indices if i bring that up i will have 81 times 10 raised to power what minus 2 okay and if you do the same for the denominator you also get that 2.25 that two point two five is the same as two two five times ten also raised to power minus two so if we substitute that here we are going to have eighty-one times ten raised to the power minus two but there is already 10 raised to the power minus 5 here and that's going to give me for the denominator 2 to 5 times 10 raised power minus 2 times 10 raised to the power 7 okay so uh by the time i apply the multiplication rule for the numerator and the denominator that is going to give us 81 times 10 raised to the power minus 2 plus minus five which is minus seven all over two to five times uh ten raised to the power minus two plus five and sorry plus seven and that is going to give us uh sorry minor yeah that's going to give us my positive five and so we'll have positive five there and then now to be able to simplify all of this what do we do you remember that this is the same thing as the square root of um 81 over 225 you know multiplied by the square root of 10 raised to the power minus 7 all over 10 raised to the power 5. now because of course you know that i can split this division here if i bring this thing back again by third rule i will still get back by 81 times the numerator here two to five times the denominator here okay so and if that is true i know that square root can go on numerator and denominator and that will give me 9 over 15 because the square root of 81 is 9 and the square root of 2 to 5 is 15 by now this is the same thing as now of course we know that square roots by the the fractional power that we just talked about is the same thing as raised to the power half and so i'm going to get 10 raised power minus 7 over 10 raised to the power 5 or raised to the power half now which is the same thing as 9 over 15 is 0.6 multiplied by now by division rule this is going to give me 10 raised to the power minus 7 minus 5 which is minus 12 or raised to the power half then my power power rule this is going to be a multiplication of powers so i'll have there a 10 now mu i before then uh by the standard form i just talked about 10 sorry 0.6 is the same as 10 6 times 10 raised to the power minus 1 and that is this one is done when you open up this bracket you will get 10 raised to the power -6 and then when you now apply the multiplication rule here you have 6 times 10 raised to the power minus 7 now so that is the solution to the a part of this problem here which i said i'm going to just take care of together with you that the b part is going to be an exercise for you which i expect you to do and i want to see your solution in the comment section below so please click the comment section take time pause this video and solve this particular problem and let me see your solution in the comment section right that will be the end of our lesson for today please try to subscribe the subscription is an encouragement for us it makes us more visible and so i covet your subscription and also please give us thumbs up of course to encourage us and then also put up your comments in the comment section below thank you and see you in our next video bye