Sums of powers

Okay, so imagine you have three blocks. Each block is a different color: red, blue, and yellow. Now, let's say you want to add up the numbers 1, 2, and 3. You could just count them and say 1 + 2 + 3 = 6, right?

Well, what if you want to add up some numbers that are all raised to the same power? That means you take each number and multiply it by itself (for example, 2 raised to the power of 3 is 2 x 2 x 2 = 8). This is called a sum of powers.

Let's use our blocks to understand this. First, we need to choose a power. Let's say we want to add up the numbers 2, 3, and 4, but we want to raise each number to the second power (that means we'll multiply each number by itself).

So, we'll take the red block (which is 2), and multiply it by itself. 2 raised to the power of 2 is 2 x 2 = 4. Then we'll do the same thing with the blue block (which is 3): 3 raised to the power of 2 is 3 x 3 = 9. Finally, we'll do it with the yellow block (which is 4): 4 raised to the power of 2 is 4 x 4 = 16.

Now we can add up those three numbers (4 + 9 + 16) to get our sum of powers: 29. And there you have it! A sum of powers is just when you add up a bunch of numbers that have been raised to the same power.