today we're going to look at indices and index laws at National five level so let's go for the whole philosophies and then every single past paper I'll give you an in-depth teaching them indices first before we get to the past papers but let's get started what are the laws of indices or indices are powers right so if I have say e to the power of M Times by a to the power of n a and then there's been any numbers I can just add the power so that's law one e to the power of n plus n so for example that would be ever had a cubed times a squared answer would be e to the power of five notice it has to be a it cannot be a different thing here and here it can be a cubed times B squared that doesn't give you the rule okay has to be exactly the same and that works with numbers as well it could be three cubed times three squared that'd be three to the power five you get a similar low if you divide so if I do a to the power of M divided by a to the power of n I can just take away the powers e to the power of M minus in and that works if it's fractions as well because fraction means divide so where's law one there's law two that's a good two main ones but then you've got a bunch of other ones you need to be aware of we've got the power paragraph So if I had a to the power of M and then all to the power of n I can just time the powers Let's do an example of that one if I had 3 squared and then cubed that'd be two freezes six that'd be free to a pair of six another example of this one though you need to be very careful is sometimes you'll combine a number with a letter such as three a squared and then to the power of four well that means you need to split that up into three to the power of four and then it's e to the power of two fours is a I would have to work out three to the power 4 which is eight to one so you will get e to one e to the a so be aware of that one and then low four or a law of is always over anything to the power of zero is equal to one anything to five zeros one five suppose Z was one seventy power zero one banana step over zeros one anything I talked about of one let's take a negative index so let's say we had a to the minus M we can rewrite that as one over a to the m or conversely if I had one over a to the m I can write it as a negative power okay you need to be aware of the ones that come up in the context the questions where actually leave your answer as a positive power okay the last two you can combine it and what I'm going to do is to is when you've got fractional Powers okay so let's look at e to the power of one over n what you know what that means is you've got the nth root of a and that's not sex so let's take an example for that one shall we a simple one would be e to the half that means you've got this square root of a another example would be e to the quarter that means you've got the fourth root of a you're gonna be able to go back and forth between those two forms and then taking any fraction and oh what's when your number one Top's not one let's say a to the power of M over n then we can write that as the nth power the bottom the denominator as the the N through of a to the m the a to the m just comes along for the right so let's take an example of that let's say we had a to the power of three quarters that would be the fourth root of a cubed now often in National five you'll ask the evaluated with a number to a fractional power let's take a simple example let's say we had eight to the power of two farts well that means the cube root of a squared and there's two ways to do this you go to the V squared first which is 64 then cubic which you may not know or you can cube root the eight first and then just Square the answer however we're going to give you the cube root of eight is two so that's just 2 squared still which would be four let's look at some specific past paper questions now okay sample past paper question here this is from 2014 paper to question eight so it's a calculator question even though you probably don't need to it so you've got simplify n to the power of 5 times 10 and over two N squared so that is the same as n to the power over five times ten and well that's a power one over 2N squared so let's just deal with the top and bottom separately so doing the top we've got 10 and 6. add the powers over 2 and squared 10 divided by 2 is 5 and and I can take away the powers now six minus two is four okay laws of indices five mass 2015 people one question fourteen evaluate 8 to the power of Five Thirds so it's a non-calcular equation so we're gonna have to change that in our format we can work on so e to the power of Five Thirds formal laws of indices that is our root and what type of group is that well the bottom number tells us cube root and the E and the five go along under the figure so it's the cube root of eight to the power of five so I can either cubic it first ones after the fact might as well do it first because 80.5 is a hard sum so the cube root of eight is two times two times two is eight so that gives me 2 to the power of five still because it's just a cube root of eight I've done two to the power of five that's two times two is four times two is eight times two is sixteen times two is 32 and we're done there but 16 paper 2 question 10 simplify N squared cubed times ten to the main n to the minus ten give your answer as a positive power so let's start with the N squared cubed that gives me two times three is six times n to the minus ten times and indices which means we can add the powers so that gives me n 6 take away ten is negative four leave your answers opposed to power the last one over n to before and one laws of indices Screen S5 Mass 2017 paper 2 question 12 explains one over the cube root of x in the form x to the power of n so we've got 1 over cube root of x you need to know from your laws of indices that a cube root can be written as x to the power of a fraction the fraction is one over three free from the cube root and because X has got like a management one here once we've done that your second law is if you've got a fraction you can take a make fraction disappear take that up to the top and that becomes a negative power so in other words we get x to the minus one for final answer lots of indices sqa National five maths 2018 paper one question 15 remove the brackets and simplify so we've got two thirds P to the four squared so this is your power power rule so remember we looked at this already we can take the two thirds and that gets squared as well and then the P has to so the four has to be squared but we can use the power paragraph four times two is eight so it simplifies to that to start with we only need to work at what two four squared is well that's two thirds times two-fourths so I could Square the top which is four and square the bottom which is nine and I've got P to the eight and we're done there is fraction is simplified okay loads of indices x three and that's a five hours 2019 paper 2 questions 16. simplify e to the 4 times 3A all over the square root of a so let's just write that out e to the four times three a all divided by the square root of a well let's deal with the top first then three is just Times by one which is three and then the powers I can put a little one on S and add the powers log 1 4 plus 1 is 5. so I get eight to the five and that's over the square root of a now to divide by the square root of a I'm doing indices I need everything to be indices so that's going to be 3 8 to the five square root of a is e to the half and now I'm dividing indices so I can take away the powers so that is 3 to the e and it's going to be five minus a half is what I need to do now five minus a half I want to own fractions so I will change that to three a and you make novice already but five is ten halves minus one half to be very clear so that gives you free eighth and nine halves and you can leave your answer like that because it's just simplified this you put it back in the root but if you were to put it back in there it would be three times the square root of a to the power of nine would be your final answer close one people one question 15 evaluate sixteen to the power of three halves so if you have to evaluate a number to the power of a fraction you need to change that fraction into what it actually means you can work it out because as a paper one non-calculator so I've got 16 to the power of three halves you just need to know that that means the denominator tells you it's a square root of 16 and the numerator tells you you're going to cube it and remember you can Cube over now or you can Cube your answer which is probably easier it's basically when it's known calculator the square root is 16 is 4 because 4 times 4 16 so that just gives me 4 cubed 4 cubed well that's four times four is sixteen sixteen times four is 64. I want that there question 11. simplify m to the minus 2 to the power of four times M2 minus five give your answer with a positive power watch out final bet so you've got m to the minus two times four that's m to the minus eight times m to the minus five so add the powers minus eight plus minus five is minus 13. and I'm almost done that's one over m to the 13 using the fractional rule for negative indices laws of indices this we lasted by Mass 23 people one question 12 simplify five C to the power of negative 2 all over C cubed times C to the four it says give your answer with a positive power we'll deal with that last so we've got five C to the minus two or wolver well C cubed times C4 Louis one add the powers three plus four is seven C seven and now we've got wheel two a divide means we can take away the powers notice that five just ha comes along for the right so we'll just leave a five alone and then we're going to do minus two take away seven well that's negative nine so C to the negative nine no power on the five and then the last row we're going to look at is we've got a negative power you can write that as one over positive power so you could write that as five times well our times same for you one over C to the nine simplifying that that's just five over C to the nine then and one done there is an index laws at National five level hopefully you find that teaching useful and helpful if so give it a like And subscribe and considering coming back we'll do National five videos now every single week take care stay safe and goodbye