If a room has a volume of 1.15 x 10 8 cm 3 , what is the volume in km 3 ?

Now for this practice question, it says, if a room has a volume of 1.15 times 10 to the 8 centimeters cubed, what is the volume in kilometers cubed? Alright. So this is a little bit different from what we're accustomed to dealing with, because here we're not only dealing with metric prefixes, but we're also dealing with exponents that come with those metric prefixes. We're gonna approach it in the same way initially. So we're gonna have here 1.15 times 10 to the 8, centi is our metric prefix, Meters cubed. Remember, what we do here is we're going to go from the metric prefix to our base unit, and then from our base unit to our new metric prefix. So we wanna get rid of centimeters cube, so we put centimeters here on the bottom, and meters here on the top. The coefficient of 1 goes on the side with the metric prefix. So 1 centi is 10 to the negative 2. Now here's the thing, this is centimeters, but we need to cancel out centimeters cubed. So what we do here is we're gonna cube the whole thing, and we'll see what effect that has later on. Now, technically, this is centimeters cubed on the bottom, which cancels out with these centimeters cubed up top. Now we're at meters cubed. We need to get rid of meters, so put meters on the bottom, And then we need to get to kilometers, our new metric prefix, so kilometers goes on top. Remember, coefficient of 1 goes on the side with the metric prefix. So 1 kilo is 10 to the 3, and we'd have to cube this whole thing. So here meters cubes cancel out and now we have kilometers cubed. But what effect does this cubing have? Well, if we were to rewrite this, this would become 1.15 times 10 to the 8 centimeters cubed. And then we know that this Okay. So then we're gonna say next, we know that, oops. We know we have centimeters cubed here, and we know this is kilometers cubed and meters cubed. Here we have 10 to the negative 2. When you raise an exponent to an additional power, so it's negative 2 to the 3 here, you're going to multiply those. So 10 to the negative 2 cubed, that means you're going to multiply these exponents together. So negative 2 times 3 gives us negative 6. So this is going to become 10 to the negative 6 meters cubed. When you cube 1, that's just 1 cubed, that's just still 1. So this is still 1 here. So 1 centimeters cubed is equal to 10 to the negative 6 meters cubed. Alright. Now here we have 10 to the 3 cubed, so that's 3 times 3, so that's 10 to the 9. So those are the numbers that we get now after we've raised them to the 3rd power. Alright. Now, we just have to plug things into our calculator. So remember centimeters, cubes cancel out, meters, cubes cancel out, and we'll be left with kilometers cubed at the end. When you have things to a power you should put them in parentheses in your calculator, otherwise you may get the wrong answer. So now just multiply 1.5 times 10 to the negative 10 to the positive 8 times 10 to the negative 6 divided by 10 to the 9. If you do that correctly, you should get 1.15 times 10 to the negative 7, negative 7 kilometers cubed. Kilometers cubed as your final answer. So just realize here that this is similar in some ways, we just have to go from 1 metric prefix to a new one. So remember the order is metric prefix to base unit to new metric prefix. Because here we're dealing with the cube, the exponent of 3, we have to do something new and different. So this is the process you'd have to do in order to cancel out those units. So take to heart all the different types of metric prefix multipliers, how they relate to one another, and the process of going from 1 metric prefix to a new metric prefix.