Video: Comparing the Sizes of Numbers Written in Scientific Notation

Which is a greater number of objects, 2 × 10³ objects or 3 × 10² objects?

03:48

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.

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