Video Transcript
In an experiment using a normal microscope, the length of the smallest feature that can be seen is 0.2092 micrometers. Which one of the following is this value expressed in meters? 292 times 10 to the negative three meters, 2.092 times 10 to the negative three meters, 20.092 times 10 to the negative seven meters, 2.092 times 10 to the negative seven meters.
This question is asking us to relate micrometers to meters. Looking at this word micrometer, we see that the second half of the word is the word meter, which is just the base unit that we’re looking for for our final answer. The first half of the word “micro-” is a prefix attached to the base unit. We now recall that when we attach a prefix to a base unit, the prefix stands for a power of 10. In particular, the prefix micro-, which is relevant for our question, stands for 10 to the negative six. If we use the letter m to represent meters and the Greek letter 𝜇 to represent the prefix micro-, so 𝜇m is micrometers, we can write the equation one micrometer is equal to 10 to the negative six meters.
What this equation means is that we can always exchange micrometers for 10 to the negative six meters and vice versa without changing the value of our quantity. So 0.2092 micrometers is equal to 0.2092 times 10 to the negative six meters. This value has units of meters, which is exactly what we are looking for. We just need to reexpress 0.2092 times 10 to the negative six so that it matches one of our answer choices. Note that all of the answer choices have powers of 10 that are not 10 to the negative six. So we need to change our power of 10 to match one of these numbers.
Increasing the exponent of a power of 10 by one is the same thing as multiplying by 10. For example, 10 to the two is 100, which is 10 times 10 to the one. Similarly, decreasing the exponent by one is the same thing as dividing by 10. However, we need the value of our number to stay constant. So if we’re going to be dividing by 10 because we are decreasing the value of the exponent, then we also need to multiply the number by 10 elsewhere. And we can do this by moving the decimal place to the right.
For example, multiplying 2.11 by 10 gives us 21.1, which is the same number with the decimal point moved one place to the right. Similarly, if we are multiplying a number by 10 because we are increasing the value of the exponent, we need to divide by 10 elsewhere, which we can do by moving the decimal point to the left.
What’s important to realize is that changing the value of the exponent and moving the decimal point doesn’t change the ordering of these digits. This means that we can immediately eliminate choice (A) — because 292 doesn’t have a zero between the two and the nine, but our number 0.2092 does — and also choice (C) — because 20.092 has two zeros between the two and the nine, but our number only has one. This leaves choices (B) and (D). Both (B) and (D) have the same number 2.092. And 2.092 is exactly the same as our number but with the decimal point shifted one place to the right.
Now, recall that shifting the decimal point one place to the right is the same as multiplying the number by 10. And in order to make sure that this multiplication doesn’t change our overall value, we need to effectively divide the number by 10, which we can do by subtracting one from the exponent. Now, as we know, because we are dealing with micrometers, our exponent is negative six. And negative six minus one is negative seven, which is the exponent of choice (D). And we find that 0.2092 micrometers is 2.092 times 10 to the negative seven meters.
