There are 225000 atoms in a dust cloud in deep space. What is the number of atoms in the dust cloud, expressed in scientific notation to two decimal places?
Okay, so in this question, we’ve been told that there’s a dust cloud in deep space that contains 225000 atoms. Now the question might seem a bit strange. It’s asking us to find the number of atoms in the dust cloud, which we already know to be 225000. However, remember that we have to express our answer in scientific notation and to two decimal places. And the number 225000 written in this way is neither in scientific notation nor to two decimal places. So let’s first start by recalling what we mean when we say scientific notation.
Now, scientific notation is just a way of displaying any particular number. And we can write every single number as 𝑎 times 10 to the power of 𝑏, where 𝑎 is any value between one and 10. But note that the value of 𝑎 can also be equal to one, but cannot be equal to 10 and 𝑏 is any value that’s an integer, a whole number. So what we’re trying to do in this question is to write the value 225000 in scientific notation as 𝑎 times 10 to the power of 𝑏. And to be able to do that, we need to find the values of 𝑎 and 𝑏 in this particular case.
Now, the way that we can go about doing this is the following. We can first take our numerical value 225000 and multiply it by the fraction 10 divided by 10. Because, remember, 10 divided by 10 is equal to one. And we can multiply any value by one and still have the exact same value. And therefore, we can say that 225000 multiplied by 10 over 10 is still equal to 225000. So what’s the whole point of this?
Well, when we multiply 225000 by 10 divided by 10, we can write this in a slightly different way. We can combine 225000 with the denominator of the fraction. And we can write the times 10 separately. In other words, what we can do is to say that our original value 225000 is equal to 225000 divided by 10 times 10. Now, 225000 divided by 10 is equal to 22500. And so, what we’ve done then is to take our original value 225000 and write it in a form that starts to look a little bit like this. We’ve now got some number multiplied by 10 to some power. In this case, the power is one because 10 by itself is equal to 10 to the power of one.
So can we say that this is the answer to our question? Well, no, because this is not yet scientific notation. We’ve got some value multiplied by 10 to the power of some integer. But this value is not yet between one and 10. Remember our value of 𝑎 needs to be between one and 10. And so, we simply repeat our initial process once again. We once again multiplied the value on the left-hand side by 10 divided by 10. And then once again, we take the denominator of this fraction and shift it over to the value in blue. And so, what we’re left with is 22500 divided by 10 multiplied by 10 to the power of one, which we already had, multiplied by 10. Then, we can work out the value of this fraction. And we can combine times 10 to the power of one times 10. And in this second part, all we’re doing is multiplying. 10 by 10 which is equal to 100 or 10 squared, 10 to the power of two.
Looking back at the very first part, the fraction, we can say that 22500 divided by 10 is equal to 2250. So our original value 225000 is equal to 2250 times 10 to the power of two. But this still isn’t scientific notation. We need to repeat our process over and over until we get this value to be between one and 10. So multiplying once again by 10 divided by 10, we find that our left-hand side becomes 2250 divided by 10 multiplied by 10 squared times 10. And that gives us 225 multiplied by 10 to the power of three. Still not quite there yet. So multiplying it by another factor of 10 divided by 10 and doing the usual process, which gives us 22.5 multiplied by 10 to the power of four. And we’re nearly there.
In this process, we can see that this number written in blue has been constantly getting smaller. And to compensate, we’ve been increasing the power of 10 every single time. This way, this whole value is still equal to our original value, 225000. So repeating our process for one final time which looks something like this, we get a value of 2.25 times 10 to the power of five. And once again, that is still equal to 225000. It’s just a different way of writing 225000.
At which point, we see that this value is in scientific notation. Our value of 𝑎 is equal to 2.25 which is a value between one and 10 or more specifically 𝑎 is greater than or equal to one and less than 10. And our value of 𝑏 is five in this case. And five is an integer. It’s a whole number. So we found our answer in scientific notation. In other words, we can say that 225000 written in scientific notation is equal to 2.25 times 10 to the power of five. And we’ve got the added benefit that 2.25 happens to be to two decimal places. Therefore, we’ve now expressed our answer in scientific notation to two decimal places.
Now, in reality, we wouldn’t go through the whole process of multiplying by 10 and dividing by 10 every single time. Instead, what we can do is to start with our original value 225000 and realize that in this value our decimal point is positioned here. We should also recall that every time we divide our value by 10, our decimal point will shift towards the left. And so, in order to get this value 225000 to be between one and 10, we need to divide by 10 one, two, three, four, five times. And so, we can say we can divide by 10 to the power of five. But then to keep it the exact same value, we also multiply by 10 to the power of five.
Then, as we’ve already seen, 225000 divided by 10 to the power five ends up being 2.25. And so, our value in scientific notation is 2.25 times 10 to the power of five, which also happens to be the final answer to our question.