Orders of magnitude exercise example 2

Let's do a few more examples from the orders of magnitude exercise. Earth is approximately 1 times 10 to the seventh meters in diameter. Which of the following could be Earth's diameter? So this is just an approximation. It's an estimate. And they're saying, which of these, if I wanted to estimate it, would be close or would be 1 times 10 to the seventh? And the key here is to realize that 1 times 10 to the seventh is the same thing as one followed by seven zeroes. One, two, three, four, five, six, seven. Let me put some commas here so we make it a little bit more readable. Or another way of talking about it is that it is, 1 times 10 to the seventh, is the same thing as 10 million. So which of these, if we were to really roughly estimate, we would go to 10 million. Well, this right over here is 1.271 million, or 1,271,543. If I were to really roughly estimate it, I might go to one million, but I'm not going to go to 10 million. So I'd rule that out. This is 12,715,430. If I were to roughly estimate this, well, yeah. I would go to 10 million. 10 million is if I wanted really just one digit to represent it, if I were write this in scientific notation. This right over here is 1.271543 times 10 to the seventh. Let me write that down. 12,715,430. If I were to write this in scientific notation as 1.271543 times 10 to the seventh. And when you write it this way, you say, hey, well, yeah, if I was to really estimate this and get pretty rough with it, and I just rounded this down, I would make this 1 times 10 to the seventh. So this really does look like our best choice. Now let me just verify. Well, this right over here, if I were to write it, I would go to 100 million, or 1 times 10 to the eighth. That's way too big. And this, if I were to write it, I would go to a billion, or 1 times 10 to the ninth. So that's also too big. So once again, this feels like the best answer. Now let's try a couple more. So here we're asked, how many times larger is 7 times 10 to the fifth than 1 times 10 to the fourth? Well, we could just divide to think about that. So 7 times 10 to the fifth divided by 1 times 10 to the fourth. Well, this is the same thing as 7 over 1 times 10 to the fifth over 10 to the fourth, which is just going to be equal to-- well, 7 divided by 1 is 7. And 10 to the fifth, that's multiplying five 10's. And then you're dividing by four 10's. You're going to have one 10 left over. Or, if you remember your exponent properties, this would be the same thing as 10 to the 5 minus 4 power, or 10 to the first power. So this right over here, all of this business, is going to simplify to 10 to the first, or I could actually write it this way. This would be the same thing as 10 to the 5 minus 4, which is equal to 10 to the 1, which is just equal to 10. So this is 7 times 10, which is equal to 70. So 7 times 10 to the fifth is 70 times larger than 1 times 10 to the fourth. Let's do one more. So here, they're asking us 3 times 10 to the ninth is 30,000 times larger than what number? So once again, we can divide. So we have 3 times 10 to the ninth is 30,000 times larger than what number? So let's just divide by 30,000 and see what we get. And here we've written something in kind of an exponential notation, or we should say scientific notation actually. And here, we just wrote the number out. So one way we could do it is we could either write this number out and then divide, or we could write this in scientific notation. So let's do it either way. So if we were to expand the top number out, we could write that as 3 followed by nine zeros. One, two, three, four, five, six, seven, eight, nine. Let me put some commas here to make it readable. And then we're dividing that by 3 followed by four zeros. One, two, three, four. And then we could cancel out the zeros. We could say, OK, let's divide the top and the bottom by 10. Let's divide it by another 10, by another 10, by another 10. And then, let's see, we've done all the dividing by 10. And now let's divide the top and the bottom by 3. So this would become a 1. This would become a 1. So on the bottom, we're just left with a 1. And we'd have 1 followed by one, two, three, four, five zeros. So this would be 1 followed by one, two, three, four, five zeros, or 100,000. Now let's write it, both of these, in scientific notation. So 3 times 10 to the ninth, I'm just going to rewrite that as 3 times 10 to the ninth. And we're dividing that by 30,000, which is the exact same thing as 3 times 10 to the-- we have one, two, three, four zeros here. 3 times 10 to the fourth. Or I guess I really should say, we have four places after the three. So one, two, three, four. So 3 times 10 to the fourth. And so we could divide the 3 by the 3, and then that will simplify. So 3 divided by 3 is just 1. And then 10 to the ninth divided by 10 to the fourth, well that's going to be 10 to the 9 minus 4, 10 to the fifth. So it's going to be 1 times 10 to the fifth, which, once again, is 1 followed by five zeros, or the exact same thing as 100,000. So it's 30,000 times larger than 100,000.