Show that 337 can be written as the sum of a power of four and a square number.
So we want to show that 337 can be written as the sum of a power of four and a square number. So we can go through- so we can go through each of the powers of four and then see how much is left to get to 337 and if that number that’s left that we needed to add in order to get to 337 is a square number or not.
So if we’re essentially trying to decide if that number that’s left is in fact a square number, we can look at it this way. Subtracting both sides of the equation by a power of four, we have what the square number should be equal to: 337 minus a power of four. And then we will decide is that a square number.
Let’s begin with four to the power of one. That would be four. So 337 minus four would be equal to 333. And this is not a square number. So let’s try four to the power of two. Four squared is 16. 337 minus 16 is 321. This is also not a perfect square.
You can check if these numbers are perfect squares by taking their square roots in a calculator and see if we get a nice pretty natural number, such as one, two, three, four, five, and so on.
So now let’s try four to the power of three, four cubed. Four cubed is equal to 64. 337 minus 64 is 273, which is not a square number. Four to the power of four is 256. And if we take that away from 337, we get 81. And 81 is a square number. The square root of 81 is nine. Or another way to think about that is that nine squared is equal to 81.
So what this means is 337 is equal to four to the power of four plus the square of nine, a square number. But what about the powers that are higher than four, four to the power of five? Well, four to the power of five is 1024. Well, that number is way too big. We can’t add a power of a number to 1024 and get 337. Therefore, our final answer will be four to the power of four plus nine squared.