Video Transcript
Which of the following represents 0.025 milliseconds in seconds? The answer should be expressed in scientific notation so that the value contains no zero digits. (A) 0.25 times 10 to the third seconds. (B) 25 times 10 to the second seconds. (C) 2.5 times 10 to the negative three seconds. (D) 250 times 10 to the negative five seconds. (E) 2.5 times 10 to the negative five seconds.
Before we do anything, we can immediately eliminate choices (A) and (D) because both of them contain zero digits in their value. But we’re told to express our answer so that the value contains no zero digits. With that out of the way, let’s focus on what this question is asking. We’re asked to convert 0.025 milliseconds into seconds. Note that the word “milliseconds” is a prefix milli- affixed to the base unit seconds. A prefix attached to a base unit like this gives us another unit that is just some multiple of the original unit.
Our prefix here is milli-, and milli- stands for a factor of 10 to the negative three. So we can write that milliseconds are equal to 10 to the negative three seconds. And by “equal to” we mean that we can interchange one millisecond with 10 to the negative three seconds and vice versa. With this in mind, for this question, we need to convert 0.025 milliseconds into seconds. And we can write 0.025 milliseconds, where we have used ms to abbreviate milliseconds, is equal to 0.025 times 10 to the negative three seconds, where we have replaced milliseconds with 10 to the negative three seconds. This answer is represented as a mantissa times a power of 10, which is good because we need our answer to be in scientific notation. However, we do still need to remove the zero digits.
If we could somehow move the decimal point two places to the right, we would have no more zero digits because our value would be 2.5. Multiplying by 10 gives us a number which is the same as the original number but with the decimal place moved one to the right. So, to move the decimal place two places to the right, we would have to multiply by 10 and then by 10 again, and 10 times 10 is 100. So, if we multiply 0.025 by 100, we eliminate the zero digits. However, in order not to change the value, we also need to divide part of this number by 100 so that we are multiplying by 100 and also dividing by 100, the net result of which doesn’t change the overall value.
Now, the reason we’re dividing 10 to the negative three by 100 here is in part because we need to multiply 0.025 by 100 and also because 100 is a power of 10. So 10 to the negative three divided by 100 is also a power of 10, which leaves our number as a mantissa times a power of 10, which is what we need to express it in scientific notation. As we said, 0.025 times 100 is 2.5, which leaves us with 10 to the negative three divided by 10 to the two. 10 to the negative three divided by 10 to the two is 10 to the negative five. So 0.025 milliseconds is equal to 0.025 times 10 to the negative three seconds, which is the same as 2.5 times 10 to the negative five seconds.
2.5 times 10 to the negative five seconds is answer choice (E), and this is the answer that we are looking for.
