Multiplying in scientific notation example

Multiply, expressing the product in scientific notation. So let's multiply first, and then let's try to get what we have in scientific notation. Actually, before we do that, let's just even remember what it means to be in scientific notation. To be in scientific notation-- and actually, each of these numbers right here are in scientific notation. It's going to be the form a times 10 to some power, where a can be greater than or equal to 1, and it is going to be less than 10. So both of these numbers are greater than or equal to 1, and they are less than 10, and they're being multiplied by some power of 10. Let's see how we could multiply this. So this over here, this is just the exact same thing. So if I do this in magenta, this is the exact same thing as 9.1 times 10 to the sixth times 3.2 times-- actually, I don't have to write it. Let me write it all with a dot notation to make it a little bit more straightforward. I'm doing that in magenta. This is equal to 9.1 times 10 to the sixth-- let me do it in this green color-- times 3.2 times 10 to the negative 5th power. Now in multiplication, this comes from the associative property. It essentially allows us to remove these parentheses. It says, look, you can multiply like that first, or you could actually multiply these guys first. You can reassociate them. And the commutative property tells us that we can rearrange this thing right here. What I want to rearrange is I want to multiply the 9.1 times the 3.2 first and then multiply that times 10 to the sixth times 10 the negative 5. So I'm just going to rearrange this using the commutative property. This is the same thing as 9.1 times 3.2, and I'm going to reassociate. So I'm going to do these first, and then that times 10 to the sixth times 10 to the negative 5. And the reason why this is useful is that this is really easy to multiply. We have the same base here, base 10, and we're taking the product, so we can add the exponents. So this part right over here, 10 to the sixth times 10 to the negative 5, that's going to be 10 to the 6 minus 5 power, or essentially just 10 to the first power, which is really just equal to 10. And that's going to be multiplied by 9.1 times 3.2. So let me do that over here. If I have 9.1 times 3.2, so at first I'm going to ignore the decimal, so I'm just going to treat it like 91 times 32. So I have 2 times 1 is 2. 2 times 9 is 18. I'll stick a 0 here because I'm in the tens place now, multiplying everything really by 30 not just by 3. That's why my zero is there. And I multiplied 3 times 1 to get 3, and then 3 times 9 is 27. And so it is 2. So I'm adding here. 2 plus 0 is 2. 8 plus 3 is 11, carry or regroup that 1. 1 plus 1 is 2. 2 plus 7 is 9. And then I have a 2 here. So 91 times 32 is 2,912. But I didn't multiply 91 times 32. I multiplied 9.1 times 3.2. So what I want to do is count the number of digits I have behind the decimal point. I have one, two digits behind the decimal point, and so I'll have to have two digits behind the decimal point in the answer. So one, two, I'll stick the decimal right over there. This part right over here comes out to be 29.12. You might feel like we're done. This kind of looks like scientific notation. I have a number times a power of 10. But remember, this number has to be greater than or equal to 1-- which it is-- and less than 10. But this number is not less than 10. It's not in scientific notation. What we can do is let's just write this number in scientific notation, and then we can use the power of 10 part to multiply by this power of 10. 29.12, this is the same thing as 2.912. Notice, what did I do to go from there to there? I just moved the decimal to the left. Or another way to think about it, if I wanted to go from here to there, what could I do to this? Well, I would multiply it by 10. If I multiplied it by 10, I would move the decimal to the right. It would go from 2.9 to 29. So if I want to write this value, this is just this times 10. So 29.12 is the same thing as 2.912 times 10. Now, this is in scientific notation, but that's just this part. And I still have to multiply it by another 10, so times another 10. To finish up this problem, we get 2.912 times 10 times 10, or 10 to the first times 10 to the first. Well, what's that? Well, that's going to be this part right over here. That's just 10 squared. So it's 2.912 times 10 to the second power, and we are done.