 So powers and roots occupy a strange place in the order of operations. So let's consider an expression like 2 times 5 to power 3. Now definitions are the whole of mathematics, all else is commentary, so it'll be helpful to remind ourselves what we mean when we write 5 to power 3. And remember that a to power n is the same as a multiplied by itself n times. And so when we write 5 to power 3, we really mean 5 times 5 times 5. Equals means replaceable, so instead of writing 5 to power 3, I can replace it with 5 times 5 times 5. And here's the important thing to recognize, we have not performed any of the operations yet. We haven't even gotten to the question of the order in which we should perform the multiplications. Well, okay, let's do that. The basic rule of the order of operations is we go left to right unless, and here since all of our operations are multiplications, we'll go left to right. So that means we're supposed to do this 2 times 5 first, so we'll multiply, and everything else stays the same. We still go left to right, so we should multiply 10 times 5 first, and everything else stays the same. Again, we should go left to right while we're only left with one operation here, so we'll multiply 50 by 5 to get our final answer, 250. And here's an important thing to understand. This is how we should evaluate 2 times 5 to the third. 5 to the third is a product of a whole bunch of 5's, and then we can evaluate the product by going left to right. However, it's helpful to take an alternate viewpoint, so remember that we have the associative property of multiplication. For any real numbers a, b, and c, if we're multiplying them all together, we can choose which order to do the multiplication. Now, there's no reason why we'd want to do this, but if we choose to multiply the things that are similar together, and we group this product 5 times 5 times 5, we can do that first because of the associative property of multiplication, and then multiply by 2. And what you should recognize here is that we're allowed to do this. And this leads to the following idea, even though exponents are not really operations. Remember, exponents are shorthand for a repeated multiplication. But even though they're not operations, they can be treated as such, and this gives us an order of operations. And again, remember the basic rule for order of operations is arithmetic operation should be performed from left to right unless operations and parentheses are done first, then exponential expressions, then multiplication and division from left to right, and then finally addition and subtraction from left to right. So, for example, let's say we have this horrifying expression. Things in parentheses have to be done first, so let's take a look at this thing inside the parentheses. There's a fraction here, and we ought to consider the numerator and denominator of a fraction to be included inside a set of parentheses. And so that means the first thing we have to take care of is 5 minus 2. Now we have the fraction 9 thirds, and the thing to remember is that a over b can be rewritten as a divided by b. Again, still inside the parentheses, we have to take care of the 9 divided by 3 first, and then 4 plus 3, and now I have exponents and multiplication and subtraction. And even though exponents aren't really operations, we can evaluate 3 to the second first. So 3 to the second, well, that's 3 times 3, and 3 times 3 is equal to 9. So I can replace... I have 9 times 5 minus 7, multiplication goes first, and then the subtraction. A root is like an exponent. It's shorthand for a specific number. But in the order of operations, we treat it like another type of arithmetic operation. And we treat it like an exponent in the order of operations. With one important idea, there's an implied grouping symbol around the argument known as the radicand. That's the thing that's inside the root symbol. So for example, take this thing. The entire expression that's under this radical symbol, this square root symbol, we should regard that as being enclosed in a set of parentheses. Parentheses say do stuff inside first. And inside the parentheses, we have 3 to the second plus 4 to the second. Order of operation says take care of that 3 to the second first. And again, the reason we actually take care of that 3 to the second first is not because it's an exponent, but because it's a multiplication. 3 to the second is 3 times 3. Likewise, 4 to the second is 4 times 4. And so we can compute both of these values. Parentheses say do this first. So we'll add 9 plus 16. And we have the square root of 25. Now remember, square root of 25 represents a specific real number. And it's possible that this is the most we can do and we have to leave our final answer in this form. On the other hand, it's possible that we might know the square root of 25. And if we do, we can go one further step. In this case, square root of 25 is a real number that you should either know or be able to figure out. Square root of 25 is equal to... And so our final answer is 5.