Multiplying three numbers in scientific notation

We're asked to multiply 1.45 times 10 to the eighth times 9.2 times 10 to the negative 12th times 3.01 times 10 to the negative fifth and express the product in both decimal and scientific notation. So this is 1.45 times 10 to the eighth power times-- and I could just write the parentheses again like this, but I'm just going to write it as another multiplication-- times 9.2 times 10 to the negative 12th and then times 3.01 times 10 to the negative fifth. All this meant, when I wrote these parentheses times next to each other, I'm just going to multiply this expression times this expression times this expression. And since everything is involved multiplication, it actually doesn't matter what order I multiply in. And so with that in mind, I can swap the order here. This is going to be the same thing as 1.45-- that's that right there-- times 9.2 times 3.01 times 10 to the eighth-- let me do that in that purple color-- times 10 to the eighth times 10 to the negative 12th power times 10 to the negative fifth power. And this is useful because now I have all of my powers of 10 right over here. I could put parentheses around that. And I have all my non-powers of 10 right over there. And so I can simplify it. If I have the same base 10 right over here, so I can add the exponents. This is going to be 10 to the 8 minus 12 minus 5 power. And then all of this on the left-hand side-- let me get a calculator out-- I have 1.45. You could do it by hand, but this is a little bit faster and less likely to make a careless mistake-- times 9.2 times 3.01, which is equal to 40.1534. So this is equal to 40.1534. And of course, this is going to be multiplied times 10 to this thing. And so if we simplify this exponent, you get 40.1534 times 10 to the 8 minus 12 is negative 4, minus 5 is negative 9. 10 to the negative 9 power. Now you might be tempted to say that this is already in scientific notation because I have some number here times some power of 10. But this is not quite official scientific notation. And that's because in order for it to be in scientific notation, this number right over here has to be greater than or equal to 1 and less than 10. And this is, obviously, not less than 10. Essentially, for it to be in scientific notation, you want a non-zero digit right over here. And then you want your decimal and then the rest of everything else. So here-- and you want a non-zero single digit over here. Here we obviously have two digits. This is larger than 10-- or this is greater than or equal to 10. You want this thing to be less than 10 and greater than or equal to 1. So the best way to do that is to write this thing right over here in scientific notation. This is the same thing as 4.01534 times 10. And one way to think about it is to go from 40 to 4, we have to move this decimal over to the left. Moving a decimal over to the left to go from 40 to 4 you're dividing by 10. So you have to multiply by 10 so it all equals out. Divide by 10 and then multiply by 10. Or another way to write it, or another way to think about it, is 4.0 and all this stuff times 10 is going to be 40.1534. And so you're going to have 4-- all of this times 10 to the first power, that's the same thing as 10-- times this thing-- times 10 to the negative ninth power. And then once again, powers of 10, so it's 10 to the first times 10 to the negative 9 is going to be 10 to the negative eighth power. And we still have this 4.01534 times 10 to the negative 8. And now we have written it in scientific notation. Now, they wanted us to express it in both decimal and scientific notation. And when they're asking us to write it in decimal notation, they essentially want us to multiply this out, expand this out. And so the way to think about it-- write these digits out. So I have 4, 0, 1, 5, 3, 4. And if I'm just looking at this number, I start with the decimal right over here. Now, every time I divide by 10, or if I multiply by 10 to the negative 1, I'm moving this over to the left one spot. So 10 to the negative 1-- if I multiply by 10 to the negative 1, that's the same thing as dividing by 10. And so I'm moving the decimal over to the left one. Here I'm multiplying by 10 to the negative 8. Or you could say I'm dividing by 10 to the eighth power. So I'm going to want to move the decimal to the left eight times. And one way to remember it-- look, this is a very, very, very, very small number. If I multiply this, I should get a smaller number. So I should be moving the decimal to the left. If this was a positive 8, then this would be a very large number. And so if I multiply by a large power of 10, I'm going to be moving the decimal to the right. So this whole thing should evaluate to being smaller than 4.01534. So I move the decimal eight times to the left. I move it one time to the left to get it right over here. And then the next seven times, I'm just going to add 0's. So one, two, three, four, five, six, seven 0's. And I'll put a 0 in front of the decimal just to clarify it. So now I notice, if you include this digit right over here, I have a total of eight digits. I have seven 0's, and this digit gives us eight. So again, one, two, three, four, five, six, seven, eight. The best way to think about it is, I started with the decimal right here. I moved once, twice, three, four, five, six, seven, eight times. That's what multiplying times 10 to the negative 8 did for us. And I get this number right over here. And when you see a number like this, you start to appreciate why we rewrite things in scientific notation. This is much easier to-- it takes less space to write and you immediately know roughly how big this number is. This is much harder to write. You might even forget a 0 when you write it or you might add a 0. And now the person has to sit and count the 0's to figure out essentially how large--or get a rough sense of how large this thing is. It's one, two, three, four, five, six, seven 0's, and you have this digit right here. That's what gets us to that eight. But this is a much, much more complicated-looking number than the one in scientific notation.