Prime Factorisation: Indicial Equations in Multiple Bases
Transcript
Question six. Now it’s asking us to find two to power x asked to find x again basically. These are prime numbers. So I can’t do anything much with them. So I am trying to worry about my right hand side which is not a prime number. So when it’s not a prime number, I always want to stick to my tree method. 240 divided by 2, you get 120.
Now I’m going to divide by 2 again which will get 60. Now divide by 2 again, get 30. Divided by 2 again, get 15. This time I’ll divide it by 3 and I’ll get 5. Because 3 times 5 is 15. And I can’t go any further because 5 is a prime number.
So 240, it’s 2 times 2 times 2 times 2 times 3 times 5. Now if I write that as my indices, it’s going to be 2 to the power of 4 because there’s 4 lots of 2 and 3 to the power of 1 which is just 3 and there’s only one 5, so just 5. Guys compare this! This and this! We’ve got the 5, we’ve got the 3, and we’ve got the 2. So what does this power have to be? In order for this to be equal to this. These powers must be the same, won’t they? x equals to 4. This power must be the same as this power which is 4. So x equals to 4. Just comparing the indices.
Find x and y in 2 to the power of x times 3 to the power of y equals 244. Now I think you kind of got the hang of it now. We can’t do much with the left-hand side. It’s already in its prime factors. So I need to try to convert this one. Off we go to our tree. So divided by 2, you’re going to get 72. Divided by 2, you’re going to get 36. Divided by 2, you’ll get 18. Divided by 2, you get 9, and 9, you have to divide it by 3, you’ll get 3 and that’s where we stop.
144 is 2 times 2 times 2 times 2 times 3 times 3. You can see there’s one two three four lots of 2s. Two lots of 3s. I think you guys can tell me the answer. 144 is 2 to the power 4 times 3 to the power of 2 because there’s four of 2s, two lots of 3s. 144 is now changed to this. If you compare this and this together. They must be equal, right? What’s the power of x here for 2? 4! And what’s the power of 3 here which is y? it’s 2! x must be equal to this number which is 4 and y must be equal to this number which is 2. So x is 4 and y is 2.
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