Part a) Write 3.45 multiplied by 10 to the power of five as an ordinary number. Part b) Write 0.000879 in standard form.
Well, if we consider 3.45 and we consider its place values. We have three units, four tenths, and five hundredths. So now, if we multiply this by 10 to the power of five, it means that each five digits is going to move five place values to the left. And this means that our three is gonna move from the unit column to the hundred thousand column. Our four is gonna move from the tenths column to the ten thousand column. And our five is gonna move from the one hundredths column to the one thousand column. And then, we fill the other spaces before the decimal point in with zeros.
So therefore, we can say that 3.45 multiplied by 10 to the power of five is gonna be equal to 345000. But there is another shortcut we could use to solve this kind of problem. So if we take a look at the number we’ve got and then we have a look at the power here. Well, here we’ve got power of five and what this tells us is the number of digits that has to be after the three in our answer, but before the decimal point.
Then, if we take a look at what number we’ve got. We got 3.45. So that means we’re gonna have three, four, five. And then, you take two away from five to work out how many zeros we’re gonna have because we got two numbers here after the decimal point. So therefore, it’s gonna be three, four, five, and then three zeros. And this is the answer we got because we got 345000. And that is three, then four five, and then three zeros. Okay, so now, let’s move on to part b.
And in part b, we need to write 0.000879 in standard form. Well, standard form is 𝑎 multiplied by 10 to the power of 𝑛, where 𝑎 is between one and 10. So 𝑎 is greater than one and less than 10. We say 𝑎 is greater than one. It could be one. But this won’t be much easier because one multiplied by 10 to the power of 𝑛 will just be 10 to the power of 𝑛. So we wouldn’t require the one. So therefore, we need to make sure that our 𝑎 is between as we said one and 10. Well, to work out what our 𝑎 is going to be, we take a look at the digits that are nonzero in our number. And we’ve got an eight, a seven, and a nine.
Now, if we have these digits in this order, the only value that we can have that’s between one and 10 is 8.79. And that’s because 879 is not between one and 10. 87.9 is not between one and 10. And 0.879 is also not between one and 10. So we found our 𝑎 because our 𝑎 is 8.79. Then, we know that we’re gonna multiply by 10. Well, what we want to know is what is our 𝑛. What is our power of 10 going to be?
So what we need to know is how many place values to the left our eight will need to move to get to the units column. And that’s because we’ve got 8.79. So therefore, the eight in 8.79 is a unit. So what we can see here from our little sketch is that the eight is gonna travel four place values. And that’s because it’s gonna go from the ten thousandths to the thousandths to the hundredths to the tenths and then finally to the units.
So therefore, our value of 𝑛 is going to be negative four. And the reason it’s negative four is because if we were looking at 8.79 and we’re multiplying it by 10 to the power of negative four, it’s negative four because the eight would have to go the inverse. So it would have to go four place values to the right. So therefore, that’s why it would be negative. And also, we know that if our number is less than one, then therefore the 𝑛 is going to be negative. So our power is going to be negative. So therefore, we can say that 0.000879 in standard form is 8.79 multiplied by 10 to the power of negative four.