Grades 11-12 Video Solutions 2025-2025_11-12

2025_11-12_05

Video Transcription

Okay, problem 5, grade 11 and 12, 2025 math kangaroo test. Which of the following is equal to the square root of 16 to the 16th power?  We have a, 4 to the 4, b, 4 to the 8, c, 4 to the 16th, d, 8 to the 8th, and e, 16 to the 4th. Try this on your own.  All right, so, like most problems, there are sort of a number of ways we could try and go about solving this problem. And we'll go over two of them.  So, like, the first way is this, like, very, you know, to me, sort of, like, tactile way of doing this.  So, we can convert 16 to the 16 to just a bunch of twos that we're multiplying together. And to me, I think this very visually illustrates how this works.  So, 16 can be written as 2 to the 4. And then, like, to the 16th power, we would just add that extra 16. And then finally, we can convert this into 2 to the 64.  And I like this method because 2 is prime, which means it's, like, we can't divide it down anymore. And then we can just represent this 2 to the 64 as a product of 64 twos.  And, of course, we're not going to write each of these things out. But we can just write it as sort of a series like that.  And if we want something to square into that, then what we need are just half of these twos. Because we can get sort of, like, you know, cut it in half. We can obtain 32 twos.  So, 2 to the 32 times another 2 to the 32. And this is just quite literally squaring because you're taking one thing and you're multiplying it by itself again.  And by expressing all of the twos, like, out like that, it becomes, to me, just a lot more obvious. So, our answer should be something that is equivalent to 2 to the 32.  I won't spoil what it is yet. But, you know, the next item, we'll get to the answer a little bit faster. So, yeah, 16 to the 16, again, we can, like, write that.  And instead of going about it like that, what we can do is find the square root of maybe, like, a component of it. And, like, what component, you might be asking?  Well, what we should do is sort of find the square root of the base and then take that to the 16th. Which sort of looks like this.  So, 16 to the 16 to the 1 over 2 is, well, you can go in one of two directions here.  You can either go 16 to the 8 because, you know, exponents are multiplicative when they're like this. Or you can just take 16 to the 1 over 2 to the 16.  And then 16 to the 1 over 2, as we know, well, what is the square root of 16? Well, it should be 4. And then we just keep the exponent of 16 and, well, it should be 16.  So our answer choice is 4 to the 16. And if we check, 2 to the 32 is, in fact, equivalent to 4 to the 16. So, yeah, we get this equivalence here as well.  And as if you really want to check it again, 16 to the 8 is also equivalent. So all roads lead to Rome. No matter what, our answer is C, 4 to the 16th.

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