Video Transcript
Which of the following is the number of picojoules, abbreviated pJ, in a joule? (A) 10 to the power of nine, (B) 10 to the power of 11, (C) 10 to the power of 12, (D) 10 to the power of 13, (E) 10 to the power of 15.
Okay, so we’re being asked in this question to work out which of these five powers of 10 is equal to the number of picojoules in one joule.
Let’s begin by recalling that pico- or lowercase p is a unit prefix. And it means a factor of 10 to the power of negative 12. If we consider units of picojoules, then pico- is a unit prefix which modifies the base unit of joules. Since pico- means a factor of 10 to the power of negative 12, then one picojoule must be equal to 10 to the negative 12 joules.
This equation here tells us that there are 10 to the negative 12 joules in one picojoule. In this question though, we’re asked what the number of picojoules in one joule is. That is, rather than a statement that says one picojoule is equal to some number of joules, we want a statement like this that says some number of picojoules is equal to one joule. Then, the value that fits in this blank here will be our answer to this question.
To work out what that value is, we’ll begin from this equation here and we’ll multiply both sides of it by 10 to the power of 12. This gives us that 10 to the power of 12 picojoules is equal to 10 to the negative 12 multiplied by 10 to the 12 joules.
Let’s recall that when we multiply a number raised to a power by the same number raised to another power, we add together those powers. That is, 10 to the power of 𝑎 multiplied by 10 to the power of 𝑏 is equal to 10 raised to the power of 𝑎 plus 𝑏. That means that on the right-hand side of this equation, 10 to the power of negative 12 multiplied by 10 to the power of 12 is equal to 10 to the power of negative 12 plus 12. Adding together negative 12 and 12 then gives us a result of zero. And so this 10 to the power of negative 12 plus 12 is simply 10 to the power of zero.
Let’s now recall that any number raised to the power of zero is just equal to one. Notice that we’ve used 𝑥 here to represent any general number. Applying this to the right-hand side of our equation relating picojoules and joules, we know that this 10 to the power of zero is just equal to one. We have then that 10 to the power of 12 picojoules is equal to one joule. This equation is telling us that the number of picojoules in one joule is equal to 10 to the power of 12.
We can notice that this value matches the answer given in option (C). We therefore choose option (C) as our answer. The number of picojoules in a joule is 10 to the power of 12.
