### Video Transcript

A bar is a unit that is defined as being equal to 10 to the power of five pascals. Convert a pressure of 0.48 bar to a pressure in pascals.

So in this question, we’re thinking about two different units, the bar and the pascal. We can see that the symbol for the bar is just the word bar written in lowercase, and the symbol for the pascal is a capital P and a lowercase a. Now both of these units are units of pressure. That means that it’s possible to convert any measurement in bar into a measurement in pascals and vice versa. In this question, we’ve been given a pressure of 0.48 bar, and we’re being asked to convert this to a pressure in pascals. In other words, we need to take this quantity of 0.48 bar of pressure and change the units so that we’re expressing it in pascals.

To help us do this, the question has told us that a bar is a unit that’s defined as being equal to 10 to the power of five pascals. We can write this information as an equation: one bar is equal to 10 to the power of five pascals. So we can see that measuring a pressure in pascals gives us a much larger number than measuring the same pressure in bar. In fact, we can see that in order to convert a certain number of bar into units of pascals, we would need to multiply that number by 10 to the power of five. We can apply this idea to the pressure given the question. We know that 0.48 bar must be equal to some number of pascals since they’re both units of pressure. And in order to work out what this number is, we just need to multiply 0.48 by 10 to the power of five.

So for now we can say that 0.48 bar is equal to 0.48 times 10 to the power of five pascals. This is technically correct, and we have converted this value of 0.48 bar into units of pascals. However, this isn’t a very conventional way of expressing a quantity. Usually in physics, we express very large or very small quantities in scientific notation. This is where we express a number as a decimal between one and 10 multiplied by a power of 10. Now here we’ve expressed our value in pascals as a decimal multiplied by a power of 10. However, this decimal isn’t between one and 10. So to find our final answer, we need to change that.

To do this, we can note that 0.48 is the same as 4.8 divided by 10. In other words, 0.48 is equal to 4.8 multiplied by 10 to the power of negative one. This means that 0.48 times 10 to the power of five pascals is equal to 4.8 times 10 to the power of negative one times 10 to the power of five pascals. So we’ve now expressed this value in pascals using a decimal between one and 10. However, we’re multiplying by two different powers of 10. We can simplify this by recalling that if we have any number 𝑥 raised to a certain power, let’s call it 𝑎, and we’re multiplying this quantity by the same number 𝑥 but raised to a different power, this time 𝑏, then the result is equal to 𝑥 to the power of 𝑎 plus 𝑏.

This means that where we’ve written 10 to the power of negative one multiplied by 10 to the power of five, this is equal to 10 to the power of the first power, that’s negative one, plus the second power, five. Negative one plus five is four, which means this is equal to 10 to the power of four. So rather than 10 to the power of negative one times 10 to the power of five, we can just write 10 to the power of four. We can now see that we’ve expressed the quantity in scientific notation in pascals. So our final answer is that a pressure of 0.48 bar is equal to 4.8 times 10 to the power of four pascals.