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Question Video: Converting a Large Number into Scientific Notation Physics • 9th Grade

There are 37500 000 000 photons incident on a small section of a spacecraft. Which of the following expresses this in scientific notation so that the value contains no zero digits? [A] 37500 × 10⁶ [B] 3750 × 10⁷ [C] 37.5 × 10⁸ [D] 3.75 × 10¹⁰ [E] 3.75 × 10⁸

07:15

Video Transcript

There are 37500000000 photons incident on a small section of a spacecraft. Which of the following expresses this in scientific notation so that the value contains no zero digits? (A) 37500 times 10 to the power of six. (B) 3750 times 10 to the power of seven. (C) 37.5 times 10 to the power of eight. (D) 3.75 times 10 to the power of 10. (E) 3.75 times 10 to the power of eight.

Alright, so in this question, we’re given this big number here. That’s 37500000000. And we’re asked to express it in scientific notation so the value contains no zero digits. Let’s begin by recalling that expressing a number in scientific notation means writing it in the form 𝑎 times 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. In this way of expressing a number, we can say that 𝑎 is a value that’s multiplied by 10 raised to an exponent 𝑏.

Now, we’re being asked to express this number here, 37500000000, in scientific notation so that the value contains no zero digits. That means writing it in this form here, 𝑎 multiplied by 10 raised to the power of 𝑏, where the 𝑎 in this expression contains no digits equal to zero. If we look at the five answer options that we’re given, we can see that each of them is indeed written in the form 𝑎 multiplied by 10 to the power of 𝑏. That is, in each of these cases, we’ve got a number or value out front, which is the 𝑎 in our scientific notation expression, and this is multiplied by 10 raised to some power, where that power is the value of 𝑏.

We can notice that both answer option (A) 37500 multiplied by 10 to the power of six and answer option (B) 3750 multiplied by 10 to the seven have values that contain digits equal to zero. We can also notice that in scientific notation, this quantity 𝑎 must be greater than or equal to one and less than 10. However, both 37500 and 3750 are greater than 10. That means that we can discount the answers given in options (A) and (B) because neither of these two numbers are actually written in scientific notation.

If we now look at the answer in option (C), 37.5 multiplied by 10 to the power of eight, we can see that though this value 37.5 does not contain zero digits, it is greater than 10. So it’s not in this range, greater than or equal to one and less than 10, that we require for the value of this quantity 𝑎 when writing a number in scientific notation. That means that the number given in answer option (C) is not expressed in scientific notation. And so option (C) cannot be the correct answer.

This leaves us then with answer options (D) and (E). These numbers are 3.75 times 10 to the power of 10 and 3.75 times 10 to the power of eight, respectively. In both these cases then, the value 𝑎 is equal to 3.75, which is greater than or equal to one and less than 10. In answer option (D), we’ve got an exponent 𝑏 equal to 10, while in answer option (E), the exponent 𝑏 is equal to eight. Both of these exponents are integer values. That means then that the numbers given in answer options (D) and (E) both consist of a value that’s greater than or equal to one and less than 10 multiplied by 10 raised to an integer exponent. That is, both these two remaining answer options are numbers that are expressed in scientific notation.

To work out which of these numbers in scientific notation is equal to this number that we’re given in the question, let’s recall that 10 raised to the power of 𝑏 means 𝑏 factors of 10 multiplied together. So, for example, 10 to the power of two, or 10 squared, is equal to 10 times 10. That’s two factors of 10. Similarly, 10 to the power of three, or 10 cubed, is equal to 10 times 10 times 10, which is three factors of 10.

In answer option (D), we’ve got 10 raised to the power of 10. So that’s 10 factors of 10 multiplied together. Meanwhile, in option (E), we have 10 to the power of eight, which is eight factors of 10. When we multiply a number by 10, that’s equivalent to moving the decimal point one space to the right. For example, if we take the number 3.75 and we multiply it by 10, we get a result of 37.5. And notice that that’s exactly the same result that we get if we take the decimal point and move it over one space to the right. We see that when we do this, 3.75 becomes 37.5.

So, if multiplying by 10 is equivalent to moving the decimal point one space to the right and 10 to the power of 𝑏 is just 𝑏 factors of 10 multiplied together, then multiplying by 10 to the power of 𝑏 is just multiplying by 10 𝑏 times. So that’s moving the decimal point 𝑏 spaces to the right.

Let’s now clear some space on the board so that we can apply this logic to write out the numbers given in answer options (D) and (E) as single values. In answer option (D), we have 3.75 multiplied by 10 to the power of 10. So that means that we take the value 3.75 and move the decimal point 10 places to the right. Of course, if we move the decimal point so far to the right that it goes past the end of the digits that we’ve got, then we need to be careful to add in all the zeros we’ll have before the new position of the decimal point. When we add in those zeros, we get this number here.

Writing this out a little more clearly, we can see that this is equal to 37500000000. We can then of course do the same for the number in answer option (E), which is 3.75 multiplied by 10 to the power of eight. This time, since we’re multiplying by 10 to the power of eight, we take the number 3.75 and move the decimal point eight places over to the right. Then, when we add in all the zeros to the left of the new position of the decimal point, we get this number here. Again, we can write this out more clearly, and we see that it’s equal to 375 million.

Now, let’s notice that the number we’re given in the question is equal to 375 followed by one, two, three, four, five, six, seven, eight zeros. We would say this number in words as thirty-seven thousand five hundred million. Let’s notice that when we wrote out 3.75 times 10 to the power of eight, we got a number 375 followed by one, two, three, four, five, six zeros. So that’s not the same number as the one given in the question. And so that number is not expressed in scientific notation as 3.75 times 10 to the power of eight. We know then that answer option (E) is not correct.

However, when we wrote out 3.75 times 10 to the power of 10, we’ve got 375 and then one, two, three, four, five, six, seven, eight zeros. We can see then that this is indeed the same number as the one that we’re given in the question. That means that this number from the question and this number in scientific notation are in fact the same number. And so the number 3.75 times 10 to the power of 10 is indeed the number from the question expressed in scientific notation. Our answer then is given here in option (D). The number 37500000000 expressed in scientific notation is equal to 3.75 times 10 to the power of 10.

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