### Video Transcript

Which of the following is the number of nanoseconds, abbreviated ns, in a second? (A) 10 to the power of six nanoseconds. (B) 10 to the power of eight nanoseconds. (C) 10 to the power of nine nanoseconds. (D) 10 to the power of 10 nanoseconds. (E) 10 to the power of 12 nanoseconds.

We’re being asked here to work out which of these five answer options is equal to the number of nanoseconds in one second. Let’s begin by recalling that in units of nanoseconds, the part of this word nano- is a unit prefix. A unit prefix acts to modify the base unit that it’s attached to. And the specific unit prefix nano- means a factor of 10 to the power of negative nine. In this case, we’re considering units of nanoseconds. And so the unit prefix nano- is attached to the base unit of seconds. Since nano- means a factor of 10 to the power of negative nine and we’ve got base units of seconds, then we can say that one nanosecond must be equal to 10 to the negative nine multiplied by one second. We can write this more simply by noticing that 10 to the power of negative nine multiplied by one is just 10 to the power of negative nine. And so one nanosecond is equal to 10 to the negative nine seconds.

This equation here is telling us that there’s 10 to the power of negative nine seconds in one nanosecond. However, we’re not being asked for the number of seconds in one nanosecond, but rather the number of nanoseconds in one second. In other words, we want to find an expression that says one second is equal to some number of nanoseconds. To do this, let’s start from this equation here. If we take this equation, one nanosecond is equal to 10 to the power of negative nine multiplied by one second, and we divide both sides by 10 to the power of negative nine, then on the right-hand side, the 10 to the negative nine in the numerator cancels with this 10 to the negative nine in the denominator. If we then swap over the way round we write the left- and right-hand sides of the equation, we can say that one second is equal to one over 10 to the power of negative nine nanoseconds.

Now, let’s recall that whenever we have any number raised to a negative power, then that’s the same as one divided by that same number raised to the same power but positive. In this equation then, the 10 to the power of negative nine must be equal to one divided by 10 to the power of positive nine. Then, one divided by 10 to the power of negative nine must be equal to one divided by one over 10 to the power of nine. We could rewrite this fraction by multiplying it by 10 to the nine over 10 to the nine, or equivalently multiplying the numerator and denominator each by 10 to the nine. Then, in the numerator, we’d have one times 10 to the nine, which is just 10 to the nine. And in the denominator, we’d have 10 to the nine over 10 to the nine, which is simply one.

And so we can see that one divided by one over 10 to the nine is just equal to 10 to the nine. That means that this factor of one divided by 10 to the power of negative nine on the right-hand side of this equation is simply equal to 10 to the power of nine. We have then that one second is equal to 10 to the power of nine nanoseconds. This equation tells us that there are 10 to the power of nine nanoseconds in one second. We can notice that this matches the answer given here in option (C). The number of nanoseconds in a second is equal to 10 to the power of nine nanoseconds.