Video Transcript
The speed of light in vacuum is three times 10 to the power of eight meters per second. What is the speed expressed in decimal notation? Give your answer in kilometers per second. (A) 30000 kilometers per second, (B) 300000 kilometers per second, (C) three million kilometers per second, (D) 300 million kilometers per second.
The question has given us a value of three times 10 to the power of eight meters per second for the speed of light in vacuum, which we’ve labeled as 𝑐. The speed has been given to us expressed in scientific notation. That means that it’s written in the form of 𝑎 multiplied by 10 raised to the power of 𝑏, where the value 𝑎 is greater than or equal to one and less than 10 and the exponent 𝑏 is an integer. In the speed of light in vacuum that we’re given, by comparing against the general expression for a number in scientific notation, we could identify three as the value 𝑎 and eight as the exponent 𝑏.
We’re being asked to express this speed in decimal notation with units of kilometers per second. At the moment, the speed is in scientific notation with units of meters per second. This means that there’s two steps we need to take to get our answer. First, we need to convert from scientific notation to decimal notation. And then, we need to convert the units from meters per second into kilometers per second.
Let’s begin then with this first step, converting the speed of light 𝑐 into decimal notation. Recall that decimal notation means the number is expressed as a single value with no powers of 10 involved. That means that we need to find a way to write this speed that doesn’t involve this 10 to the power of eight. Let’s recall that 10 raised to the power of 𝑏 is the same as 𝑏 factors of 10 multiplied together. For example, 10 raised to the power of two, or 10 squared, is equal to 10 times 10. That’s two factors of 10, and this works out as 100.
Likewise, 10 to the power of three, or 10 cubed, is 10 times 10 times 10, which is three factors of 10. And that works out as 1000. That means that this 10 to the power of eight in the expression for the speed of light in vacuum is just eight factors of 10 multiplied together. When we multiply a number by 10, that’s equivalent to moving the decimal point one space to the right. Then, since 10 to the power of 𝑏 is just 𝑏 factors of 10 multiplied together, this means that multiplying by 10 to the power of 𝑏 is just multiplying by 10 𝑏 times. And so that’s moving the decimal point 𝑏 spaces to the right.
Let’s now apply this logic to the speed of light in vacuum 𝑐, which we’re given as three times 10 to the power of eight meters per second. We can imagine a decimal point immediately to the right of the value three. And we know that multiplying by 10 to the power of eight shifts this eight places over to the right. When we move a decimal point to the right so that it goes past the end of the digits that we’ve got, we need to be careful to fill in all the zeros that we’ll get to the left of this new position of the decimal point. When we do that, we end up with this number here. Writing this a bit more clearly, we can see that in decimal notation, we have the speed of light 𝑐 is equal to 300 million meters per second.
We’ve now completed this first step, converting the speed of light into decimal notation. Now we just need to convert it from the current units we’ve got of meters per second into units of kilometers per second. Let’s clear some space on the board to do this.
In order to convert a speed from units of meters per second into units of kilometers per second, we need to recall that the unit prefix k or kilo means a factor of 1000. This means that one kilometer is equal to 1000 meters. Since the speed in kilometers per second is giving the number of kilometers traveled per second of time and a speed in meters per second is the number of meters traveled per second of time, then if one kilometer is equal to 1000 meters, a speed of one kilometer per second must be equal to a speed of 1000 meters per second. This means that to convert a speed from units of meters per second into units of kilometers per second, we need to take the value in meters per second and divide it by a factor of 1000.
Since we’ve already found that the speed of light in vacuum is equal to 300 million meters per second, then in units of kilometers per second, it must be equal to 300 million divided by 1000 kilometers per second. This works out as a speed of 300000 kilometers per second.
Now that we’ve also completed this second step of converting the speed into units of kilometers per second, we can compare the value that we’ve got against the four possible answer options. When we do this, we find that our result matches the value given in answer option (B). The speed of light in vacuum three times 10 to the power of eight meters per second expressed in decimal notation with units of kilometers per second is equal to 300000 kilometers per second.
