Video Transcript
Earth’s continents move very slowly in a phenomenon called continental drift. Every second a continent moves by 4.271 times 10 to the negative 13 kilometers. Which of the following expresses this distance with the appropriate unit prefix so that it contains no zero digits? 4.271 nanometers, 427.1 picometers, 4.271 picometers, 4271 femtometers, 427.1 femtometers.
This question is asking us to reexpress 4.271 times 10 to the negative 13 kilometers. There are a few things that we can notice between the number we are given and our answer choices. First, the order of digits four, two, seven, one is the same in all of the answer choices. However, in the number we are given, the decimal point is between the four and the two. But in some of our answer choices, it’s between the seven and the one, and some of our answer choices don’t have a decimal point at all.
Next, we notice that the number that we are given has a power of 10, but none of our answer choices are expressed with a power of 10. Having no power of 10 is the same thing as having 10 to the zero power because 10 to the zero power is one. So we’ll write that the powers of 10 are different between the number we are given and our answer choices. Finally, the units of our number are kilometers, and the units of our answer choices are nanometers, picometers, and femtometers. Now, each of these units is related to the unit meters represented by the letter m, but they all have different prefixes, k for kilo-, n for nano-, p for pico-, and f for femto-.
Recall that for a decimal point to change locations, we need to multiply or divide by a power of 10. Additionally, we know that the powers of 10 themselves are different between the number we are given and the answer choices. So these two differences tell us that one of the things we need to do to change the number that we are given to one of the answer choices is to multiply or divide by a power of 10.
Additionally, because the digits are all the same and in the same order, we know that the only thing we have done to our number is multiplied or divided by a power of 10. We haven’t added or subtracted any numbers, and we haven’t multiplied or divided by anything other than a power of 10. This is because any operation other than multiplying or dividing by a power of 10 would change the digits or their order. We’ll summarize this by saying we have only modified the power of 10 and not the rest of the number.
Remember that we are just trying to reexpress a number, not change it. So the two expressions must be equal, which means that if we modify the power of 10, we do need to modify something else about our value. In this case, that would be the unit prefixes. Remember that unit prefixes are stand-ins for powers of 10. So when we modify the power of 10, we will also modify the unit prefixes representing powers of 10 so that the combined effect is to keep the same overall value even though the expression will be different.
The unit prefixes we are dealing with are k representing kilo-, which stands for 10 to the third; n, which represents nano-, which stands for 10 to the negative nine; p, which represents pico- and stands for 10 to the negative 12; and f, which is femto-, and this stands for 10 to the negative 15. What we need to do now is remove the power of 10 in our original number through a combination of moving the decimal place and adjusting the prefix according to these correspondences. Now, these correspondences are only relative to the base unit. So to use them, we’ll first change kilometers into meters by replacing k with 10 to the three.
With this replacement, we have 4.271 times 10 to the negative 13 times 10 to the third meters. 10 to the negative 13 times 10 to the third is 10 to the negative 10. Now, 10 to the negative 10 is not one of the powers of 10 that corresponds to one of the prefixes we have in our answer choices. However, we can add one to this exponent if we move the decimal point one space to the left. We can also subtract one if we move it one space to the right. And we can add or subtract more than one if we move the decimal point more than one space. So we need to figure out which way to move the decimal point.
The question specifies that our final answer should contain no zero digits. However, if we move the decimal point one space to the left, it will be before the four. This gives us 0.4271, which has a zero digit. So moving the decimal point to the left cannot be the answer. Instead, we need to move the decimal place to the right. If we move it to the right one space, we subtract one from our exponent, which will make 10 to the negative 10 into 10 to the negative 11. But negative 11 is also not one of the exponents that we have in our column. However, if we move the decimal point two spaces to the right, we subtract two from our exponent. And negative 10 minus two is negative 12, and negative 12 is the exponent of the power of 10 associated with the prefix pico-.
When we move the decimal point two spaces to the right, we get 427.1. So our number is now 427.1 times 10 to the negative 12 meters. Since the prefix pico- stands in for 10 to the negative 12, 10 to the negative 12 meters is just picometers. Combining this number with the appropriate unit, we see that 427.1 picometers is 4.271 times 10 to the negative 13 kilometers expressed with appropriate unit prefixes and no zero digits.
