Video Transcript
Which is a greater number of
objects, two times 10 to the power of three objects or three times 10 to the power
of two objects?
Okay, so, essentially, what this
question is asking us is which number is larger, two times 10 to the power of three
or three times 10 to the power of two. Because we’re trying to find which
is the greater number of objects, whether it’s this number of objects or this number
of objects. And so, to answer this question,
let’s write down the numbers two times 10 to the power of three and three times 10
to the power of two in full. Or, in other words, in decimal
notation. Where if we write a number, for
example, as 𝑎𝑏𝑐.𝑑𝑒, where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒 are all different numerals,
then all of this as one represents one single number. And that number is 𝑎 lots of 100
plus 𝑏 lots of 10 plus 𝑐 lots of one plus 𝑑 lots of one-tenth plus 𝑒 lots of one
hundredth.
And, for example, this number could
be 123.45. Which represents one lot of 100, so
that’s 100, plus two lots of 10, so that’s 20, plus three lots of one, so that’s
three, plus four lots of one-tenth, so that’s four-tenths or 0.4, plus five lots of
one hundredth, which is 0.05. And altogether this number
represents 123.45. And so, decimal notation is exactly
how we’re trying to write these numbers here in order to be able to compare
them. Because currently, they’ve been
written in scientific notation. And scientific notation is when a
number is written as 𝑎 multiplied by 10 to the power of 𝑏, where 𝑎 is any number
that’s greater than or equal to one and less than 10 and 𝑏 is any integer or whole
number.
So we’re trying to take these two
numbers, which have been currently written in scientific notation, and write them in
decimal notation. So let’s do that for this number
first. This number is two times 10 to the
power of three. At which point, we can recall that
the meaning of 10 to the power of three is 10 times 10 times 10. In other words, we multiply by 10
three times. That’s what this power means. And so, we can replace 10 to the
power of three by 10 times 10 times 10. And then, we can simplify the whole
thing. We can, firstly, say that two times
10 is 20. And then 20 times 10 is 200. And then 200 times 10 is 2000. And so, what we can say is that the
number two times 10 to the power of three is the same thing as 2000. At which point, we’ve converted our
original number and written it in decimal notation.
So let’s do the same thing for this
number, three times 10 to the power of two. We can start once again by saying
that 10 squared or 10 to the power of two is the same thing as 10 times 10. In other words, multiplying by 10
two times. And so, we replace 10 to the power
of two by 10 times 10. And then we can say that three
times 10 is 30. And 30 times 10 is 300. At which point, we’ve written our
number, three times 10 to the power of two, now in decimal notation. Which means that we can now more
easily compare the two numbers, two times 10 to the power of three and three times
10 to the power of two. Because we now know that two times
10 to the power of three is the same thing as 2000 and three times 10 to the power
of two is the same thing as 300.
So if we’re asked which is the
greater number of objects, two time 10 to the power of three objects or three times
10 to the power of two objects. This question is essentially asking
us which is a greater number of objects, 2000 objects or 300 objects. And we know that 2000 is a larger
number than 300. And hence, two times 10 to the
power of three objects is greater. Which is very interesting because
this number starts with a two which is smaller than this number which is a
three. And yet, two times 10 to the power
of three is greater than three times 10 to the power of two. And that all comes down to the
power of 10 that we’re dealing with. 10 to the power of three is greater
than 10 to the power of two. At which point, we’ve reached the
answer to our question.
