Prime Factorisation: Indicial Equations in Single Bases

Transcript

Okay, it’s a pretty different type of question. It says find x in 2 to the power of x equals to 32. Now in these kinds of numbers, I want to break down anything that’s not prime. So see how 32 is not a prime number. Now 2, I can’t really do anything much with this side because 2 is a prime number. I can’t actually break it down anyway, so I’m just going to try to worry about the right-hand side. See how 32 I’m going to use my truth and see if I can break it down into his prime factors and see if I get anything similar to my left-hand side.

So 32, I’m going to divide by 2, and I’ll get 16. I’m going to divide by 2 again and get 8. Divided by 2 again get 4 and divide it by 2 again get 2. And we can’t go any further than that. Now we only have 2. 2 are the only prime factor of 32. So how many 2s are there? One two three four five, there’s five lots of 2s. So I can say that 32 is 2 to the power of 5. Compare this with my left-hand side, 32 is 2 to the power of 5. So 2 to the power 5 must be equal to 2 to the power of x. So x is 5. Does that make sense guys? Hopefully, most of you got that. It says find x in see how 3 is there? The power is 2x minus 1. It’s not just x. It’s 2x minus 1 which is pretty big, isn’t it? Like bigger than what we’re used to. We used to have just power of x but that’s okay we’ll just keep it the same way nothing’s a big deal. We’re just going to as long as we keep with the same structure as what we did in the previous questions. Everything will be the same, okay? So don’t freak out.

This part 243, again let’s just break it down into our prime factors. Divided by 3 to start off with should get 81. Now I’m going to divide by 3 to get 27, divided by 3, I’ll get 9 and lastly I’ll divide it by 3 to get 3, okay? So I’ve done all that and you can see that there’s only 3s, only the prime number is 3. So there’s one two three four five lots of 3. So 243 will be 3 to the power of 5 because it’s 5 lots of 3. Compare it! 243, I broke it down into its prime factors and it became 3 to the power 5. And I want that to be equal to this.

So see how the base is the same they’re both 3. So therefore the power or the indices must also be the same. So I’m going to say that this power here must be same as this power here. 2x minus 1 must be equal to 5. And now using your algebra skills. I just need to find what x is. So this is my equation now to solve it. So move the 1 over to the side, so basically you’re adding 1, so it’s going to be 5 plus 1 which is 6, and then divide by 2 to get x. So x is 6 divided by 2 which is 3, there we go, we found x, okay? Can you see that guys?

 

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