Topic C: Applying operations with rational numbers to expressions and equations
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Simplify negative 1 times this expression in brackets, negative 7 plus 2 times 3 plus 2 minus 5, in parentheses, squared. So this is an order of operations problem. And remember, order of operations, you always want to do parentheses first. Parentheses first. Then you do exponents. Exponents. And there is an exponent in this problem right over here. Then you want to do multiplication. Multiplication and division. And then finally, you do addition and subtraction. So let's just try to tackle this as best we can. So first, let's do the parentheses. We have a 3 plus 2 here in parentheses, so we can evaluate that to be equal to 5. And let's see, we could do other things in other parts of this expression that won't affect what's going on right here in the parentheses. We have this negative 5 squared. Or I should just, we're subtracting a 5 squared. We want to do the exponent before we worry about it being subtracted. So this 5 squared over here we can rewrite as 25. And so let's not do too many steps at once. So this whole thing will simplify to negative 1. And then in brackets, we have negative 7 plus 2 times 5. And then, 2 times 5. And then close brackets. Minus 25. Now, this thing-- we want to do multiplication. You could say, hey, wait. I still have a parentheses here. Why don't I do that first? But when you just evaluate what's inside of this parentheses, you just get a negative 7. It doesn't really change anything. So we can just leave this here as a negative 7. And this expression. We do want to evaluate this whole expression before we do anything else. I mean, we could distribute this negative 1 and all of that, but let's just do straight up order of operations here. So let's evaluate this expression. We want to do multiplication before we add anything. So we get 2 times 5 right over there. 2 times 5 is 10. That is 10. So our whole expression becomes-- and normally, you wouldn't have to rewrite the expression this many times. But we're going to do it this time just to make sure no one gets confused. So it becomes negative 1 times negative 7 plus 10. Plus 10. And we close our brackets. Minus 25. Now, we can evaluate this pretty easily. Negative 7 plus 10. We're starting at negative 7. So I was going to draw a number line there. So we're starting-- let me draw a number line. So we're starting at negative 7. So the length of this line is negative 7. And then, we're adding 10 to it. We're adding 10 to it. So we're going to move 10 to the right. If we move 7 to the right, we get back to 0. And then we're going to go another 3 after that. So we're going to go 7, 8, 9, 10. So that gets us to positive 3. Another way to think about it is we are adding integers of different signs. We can view the sum as going to be the difference of the integers. And since the larger integer is positive, our answer will be positive. So you could literally just view this as 10 minus 7. 10 minus 7 is 3. So this becomes a 3. And so our entire expression becomes negative 1. Negative 1 times. And just to be clear, brackets and parentheses are really the same thing. Sometimes people will write brackets around a lot of parentheses just to make it a little bit easier to read. But they're really just the same thing as parentheses. So these brackets out here, I could just literally write them like that. And then I have a minus 25 out over here. Now, once again, you want to do multiplication or division before we do addition and subtraction. So let's multiply the negative 1 times 3 is negative 3. And now we need to subtract our 25. So negative 3 minus 25. We are adding two integers of the same sign. We're already at negative 3 and we're going to become 25 more negative than that. So you can view this as we're moving 25 more in the negative direction. Or you could view it as 3 plus 25 is 28. But we're doing it in the negative direction, so it's negative 28. So this is equal to negative 28. And we are done.