Question Video: Using Order of Operations to Evaluate Numerical Expressions Involving Exponents

Calculate (0.2)² × 4(13 + 7)² − 5².


Video Transcript

Calculate 0.2 squared multiplied by four multiplied by 13 plus seven all squared minus five squared.

Now to work out this problem, we need to know what order to do our operations in. And to help us remember, we have this. It’s something called PEMDAS. And it’s a memory aid to help us decide in what order to do our operations. So, first of all, we have parentheses. Then, we have exponents. Next is multiplication then division, addition, and then, finally, subtraction. So, we’ve got this now. And this tells us what order to do our operations in. So, now we can get on and solve the problem.

So, we start at the top. So, we start at parentheses. So, if we do the parentheses first, we’ve got 13 add seven inside our parentheses. And that’s gonna be 20. So therefore, what we have is 0.2 all squared multiplied by four multiplied by 20 all squared minus five squared. We don’t have to worry about the first set of parentheses because this was just telling us that we had 0.2 then the whole of 0.2 is gonna be squared when we do that.

So, now that we’ve dealt with the parentheses, we’re gonna move on to the exponents. And we’ve got three lots of exponents in our expression. So, we’ve got 0.2 squared, we’ve got 20 squared, and we’ve got five squared. So, first of all, we’re gonna square 0.2. And when we do that, we get 0.04. So, how do we do that?

Well, we’ve got 0.2 multiplied by 0.2 because that’s what 0.2 squared means. And when you’re multiplying decimals, one way of dealing with it is to remove the zero point and just have the value that we’ve got. So, in this case, it’s two multiplied by two, which would give us four.

And then, what you do is we go back to the original calculation and we see how many digits were there after the decimal point in the original calculation. And we can see that there are two because it’s 0.2. So, there’s the two after the decimal point, and then multiplied by 0.2, so another two after the decimal point. So, we’ve got two digits after the decimal point. So therefore, our answer has to include the same.

So, what we do is we put a zero point zero in front of our four cause that the answer we got from two multiplied by two. And we get 0.04. And this is correct because you can see here we’ve now got our two digits after the decimal point. So, that’s how we squared 0.2.

So then, we’ve got multiplied by four multiplied by 400. And we got 400 because 20 squared is 400. And again, we can think about how we calculate that. If you got 20 multiplied by 20, it’s the same as two multiplied by two which is four. And then we’ve got to think, well, how many zeros did we have in the original calculation? We’ve got two. So then, what we’ve got to do is multiply our four by 100 because it’s like multiplying it by 10 and then 10 again. So therefore, we get 400 as our answer. And then, finally, we’ve got minus 25. And that’s because five squared is 25.

Okay, great, we’ve dealt with the exponents. Now we can move on to the multiplication. Well, I’ve actually rewritten our expression to help us with this. So, we’ve now got 0.04 multiplied by four and then we had multiplied by 400, cause we had that because it was outside the parentheses. But what I’ve done here is put the multiplication sign in to make it easier for us to see what’s going on, and then minus 25.

Well, we can actually deal with the multiplication in any order we want because we got 0.04. So, we could do that multiplied by four, or four multiplied by 400, or 0.04 multiplied by 400, whichever part we want to do first. Well, I’m gonna start with 0.04 multiplied by four. And again, the same as the previous method when we’re dealing with a decimal, I’m first gonna start with four multiplied by four.

So, if we do four multiplied by before, we get 16. And then, what I can see is that there are two digits after the decimal point. So therefore, needs to be two digits after the decimal point in our answer. So, it won’t be 16. It’ll be 0.16. And that’s to give us our two digits after the decimal point.

So, once we’ve completed that first multiplication, it leaves us with 0.16 multiplied by 400 minus 25. Well, now we’re gonna do the multiplication 0.16 multiplied by 400. So, what we’re gonna do is split it up to help us out calculate it more easily. So, we’ve got 0.16 multiplied by 100 multiplied by four. That’s cause four multiplied by 100 gives us 400. And this is gonna leaves us with 16 multiplied by four. That’s because if you multiply anything by 100, then each of the digits move two place values to the left. So therefore, 0.16 becomes 16. So, we’ve got 16 multiplied by four, which is gonna equal to 64. We can work that out because we’d have four multiplied by 10 is 40. Four multiplied by six is 24. Add them together, gives us our 64.

So, now that’s our multiplication complete. And then, we’ve got no division and no addition. So, we can leave both those out and move straight onto the last part which is S, subtraction. And we’ve got that because we’re left with 64 minus 25, or 64 subtract 25. And when we carry this out, we get 39. We can do that using column subtraction.

Well, first of all, we do four minus five. But we can’t do this, so we have to borrow from the tens column. So, it turns to a five. So, now we’ve got 14 minus five which is nine. Be careful here because the most common mistake made by students is if they see four over five, they think, well, just do five take away four which is one. But this is not the case because it’s four minus five not five minus four.

Okay, now we can move on to the tens. We’ve got five minus two. So, now we’ve got 39 cause that’s what we’re looking for. And we’ve completed our PEMDAS cause we’ve got subtraction done. So, we can say that the answer to 0.2 all squared multiplied by four multiplied by 13 plus seven all squared minus five squared is 39.

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