Mistake 4 - Squaring Negatives
Students often make the mistake of squaring a negative that's not in parentheses. What will happen is that they look at it and say, "Okay, well, I've got this negative 3 squared. So that obviously equals a 9 and then for the second one, I have negative 3 squared and that equals a 9 also." The mistake in that is that the grouping symbols make a difference - a very large difference. So you have to be careful about that. So, of course, this is incorrect, whereas the second one actually checks out. Let's go through it and talk about what the difference actually is. Here's the correct way of thinking about it. This negative right here means that we’re actually having to multiply by a negative 1. If you don't see grouping symbols around the 3 or the negative 3, that 2 goes along with the 3. So any number that is being raised to a power, if there is no grouping symbols around it, then it only goes with that number beside it. So what we’re really looking at is that this negative here implies that there's a negative 1. So we have negative 1 times (and then, of course, this three squared) is just meaning 3 times 3. So we have negative 1 times 3 times 3 which gives us a negative 9. You have to be really careful about those symbols, especially the negative symbols and grouping symbols. Notice on this second one, we've got this two as a power of this negative 3 inside these grouping symbols. So that means that 2 goes along with this whole entire expression as a negative 3. In order to rewrite this, what this means is that we actually have negative 3 times negative 3. And so, negative 3 times negative 3 is just a positive 9. These are really the correct way to look at powers and things with parentheses in them.