Video instructions and help with filling out and completing Are Form 5495 Index

Instructions and Help about Are Form 5495 Index

Hello and welcome to a lesson on the rules of indices this lesson will be equally applicable to higher-tier GCSE and to the early modules available what I'm going to do is deduce the eight rules of indices by looking at examples in algebra then once I've got all the air rules of indices we look at a further ten examples to put them into practice so let's begin let's first begin by considering y squared multiplied by y cubed what does that actually mean well Y squared is y times y and Y cubed means Y times y times y so if we multiply those together the brackets actually served more useful purpose we can just remove the brackets so we've got Y times y times y times y times y but that is exactly what we mean by Y to the power 5 so Y squared multiplied by Y cubed means Y to the power 5 now that leads us to our first rule a more general rule if we've got X to the power P multiplied by X to the power Q then what that's going to give us is X to the power P plus Q because if we've got to lots of Y multiplied together and here we've got 3 lots of Y I'm what you'll add together all together we end up with 2 Plus 3 lots of Y so more generally if we have P lots of X multiplied together Q lots of X multiplied together then all together we'll have P plus Q lots of X multiplied together moving on now to find a second rule for indices let's consider y to the power 5 / y squared what do we mean by that well let me write out Y to the five first and then divide that by y squared which is y times y now what I can do now is start canceling the Y's I can cancel at Y with that Y we place them with two ones I can reconcile that Y with that Y and replace those with two ones but I've now run out of Y's on the bottom to cancel and what I've got on the top of the fraction I've got Y times y times y which is y cubed and on the bottom I've just got one times one so Y cubed divided by one is just Y cubed so now look what's happened Y to the 5 divided by Y squared is y cubed the two y's that ended up in the denominator multiplied together cancelled out two of the five I had on the top and that leads me to my second general rule if I've got X to the power P divided by X to the power Q I will end up with X to the power P minus Q because Q lots of X will cancel out Q lots of X on the top and I'll be left with P minus Q of them multiplied together for our third rule of indices let's begin by considering Y squared or cubed if we want something cubed we want the thing multiplied by itself multiplied by itself again so this is equal to Y squared multiplied by Y squared multiplied by Y squared but it's important in our rules for indices are consistent so this has to be consistent with our first rule and our first rule tells us the Y squared times y squared is y to the two - which is why to the power 4 then when we multiply Y to the power 4 by Y squared again by our first rule we'll get Y to the 4 plus 2 which is y to the 6 so for consistency we require this answer to be Y to the 6 and we can also verify that by writing everything out in full because Y squared is y x 1 the second y squared is y x y and the third y squared is y x y and if I multiply those together then what I've got is 6 lots of Y all multiplied together which again is precisely what we mean by Y to the power 6 so that leads to our third rule generalizing X to the power P all to the power Q is equal to X to the P times Q which we can write as PQ y squared all to the power 3 is y to the 2 times 3y to the 6 but 2 and 3 could be any numbers so X to the P or to the power Q is X to the P Q and we now have the most three important rules of indices the other 5 will all follow automatically from these three so let's find rule number 4 let's consider y to the 4 divided by y cubed well Y to the 4 is y times y times y times y if I divide that by Y cubed we can see that I can cancel Y with Y I can cancel Y with y again I can cancel Y with Y again and all I'm left with is Y on the top times 1 times 1 times 1 which is still y and on the bottom 1 times 1 times 1 is 1 so the answer is why so why the power 4 divided by y cubed is y but to be consistent with our second rule we have Y to the 4 divided by Y to the 3 is y to the power 4 minus 3 to be consistent with our second rule and 4 minus 3 is 1 so that has to be Y to the 1 so we have to agree that what we mean by Y to the 1 is just Y itself and that gives us our fourth rule X to the power 1 simply means X itself moving on to deduce our