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Studying for a test? Prepare with these 8 lessons on Exponents, radicals, and scientific notation.

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# Orders of magnitude exercise example 2

Video transcript

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.