When multiplying a product involving scientific notation, make sure that the "A" value is less than 10. Once you multiply and the A value is greater than 10, move the decimal so that the the value is less than 10, then change the exponents accordingly. If you needed to move the decimal to the left to make the A value less than 10, then add the exponent. If you needed to move the decimal to the right, then subtract from the exponent.
To change a number from scientific notation to standard notation, move the decimal based on the exponent of the 10. If the exponent is positive, move the decimal to the right. If the exponent is negative, move the decimal to the left.
When doing homework with the exponents a problem like this might be one of the trickiest ones that you see. Let’s take a look.
Write 8 times 7.1 times 10 to the negative 5th in scientific and standard notation. Now when you see this what you’ll be tempted to do is tempted to distribute the 8 through. Well that’s the right thing to do. But what happens is that when you multiply 8 times 7.1 you get an A value greater than 10 which makes things little bit tricky. Let’s take a look.
8 times 7.1 is 56.8 times 10 to the -5th. Well now since our A value is greater than 10 what we’re going to need to do is we’re going to need to move this decimal point one to the left. Well that’s great but what we also need to do is we need to adjust this value. Since we’re moving the decimal point one to the left we need to increase this power by one. Now it wouldn’t be 10 to the negative 6, we’d be increasing so it’d actually be 10 to the negative 4th. So if we were to write this in scientific notation what we would get is we would get 5.68 times 10 to the negative 4th.
Remember this is a little bit tricky because when we distribute our a value is 56.8 shows is greater than 10 which means it is something we can’t have. So now that we’ve written it in scientific notation let's looks at what it’d look like in standard notation. So we have 5.68 times 10 to the negative 4th. Since it’s to the negative 4 we know that need to move our decimal place 4 to the left. So if I were to add them to the zeros and then write out term we’d have to move the decimal point over, 1, 2, 3, 4 to right here. So let’s clean this up a little bit. We have 0.000568; a very, very small number.
So problems like this are really difficult and it’s important to be able to go back and forth between scientific notation and standard notation. It’s a really important skill that you’ll need to have when you’re doing your homework.