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Hi there and welcome to another video from Haiti mats it's mr. Haggerty here and in this video we're talking about index form again our sixth video but this time we're talking of powers of powers and I'll explain what I mean now before we get going what does 49 to the power of 4 mean well hopefully you know that it means the following 49 x 49 x 49 x 49 right we know that now here's the news sort of style of question what does 7 squared which by the way is 49 I picked that number on purpose to the power for me well it must mean 7 squared x 7 squared x 7 squared x 7 squared right it's got to mean that if 49 to the power of 4 means 49 multiplied by itself 4 times 7 squared to the power 4 must mean 7 squared multiplied by itself 4 times but what does 7 squared actually mean what this means 7 times 7 right and what does this one mean 7 times 7 and this one 7 times 7 and this one 7 times 7 and we're multiplying the 7 squares by each other support multiplied in there now how many 7s have we got multiplying together one two three four five six seven eight this is going to therefore be written as 7 to the power of eight now can you spot what I'm trying to show you here I've showed you that 7 squared to the power of 4 is the same thing as 7 to the power of 8 what am I trying to show you well if you are doing a power raised to another power so 7 squared the powers to all of that raised to another power the answer is 7 to the 8th you can just multiply the 4 and the 2 together to get 8 and that's it done simple as that so let's try and formalize that when simplifying powers of a power you can multiply the powers okay so let me give you three examples here seven squared to the power of four would be 7 to the 2 multiplied by 4 which will be straight 7 to the power of 8 5 cubed to the power of 7 would be 5 to the 3 multiplied by 7 which is going to be 5 to the 21 and 2 cubed to the negative 4 is going to be 2/3 multiplied by negative 4 3 multiplied by negative 4 is negative 12 and that's how you would do these questions it's as simple as that and I just want you to notice the 7 is preserved 7 squared to the power of 4 is 7 to the eighth we still have our base number of 7 5 cubed to the 7 is 5 to the 21 we still have the 5 as the base number 2 cubed to the negative 4 is 2 to the negative 12 that base number doesn't change or get erase the power of 4 or power 7 that stays the same is the powers only that multiplied by each other and it's as simple as that so let's try some questions okay we're asked to simplify the following so 4 squared to the power of 3 is 4 to the 2 multiplied by 3 which is 4 to the 6 that's right be careful not to add the powers and say that's 4 to the 5 5 to the 7 all squared will be 5 to the 7 multiplied by 2 which is 14 5 to the 7 to the negative 2 would be 5 to the 7 multiplied by negative 2 which would be 5 to the negative 14 as a positive times a negative is a negative 5 to the negative 7 multiple to the power 2 would be 5 to the negative 7 times 2 which again is 5 to the negative 14 because a negative x positive negative but the last one 5 to the negative 7 all to the negative 2 well that's going to be 5 to the negative 7 multiplied by negative 2 and negative x negative is positive so we'll get 5 to the positive 14 and we're done evaluate the following pause and have a go okay here two squared to the power of three is going to be two to the power of six now can we work out two to the power of six it's two times two times two times two times two times two two times two is four times another two is eight times another two is sixteen times another two is 32 times another two is 64 because they said evaluate we needed to give the number answer so 10 to the power of four all to the power of two is going to be 10 to the power of four multiplied by 2 which is 10 to the power of 8 and that means 10 multiplied by itself eight times so let's write that out okay and let's evaluate this 10 x 10 100 times another 10 is a thousand times 10 ten thousand times 10 a hundred thousand times 10 a million times 10 10 million times 10 a hundred million so the answer is going to be a hundred million last one 10 squared all to the negative one so that's going to be 10 to the 2 multiplied by negative 1 2 multiplied by negative 1 is negative 2 so 10 to the negative 2 and this means 1 over 10 squared or 1 over 100 and 1 over a hundred or a hundred 3 should know as 0.01 okay so let's try the next question now this skill here I haven't talked about in the video but we can do based on what we've learned is going to be so useful for the future and hard questions and future

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