In this video, we take a look at what happens when we take a number in scientific notation and raise it to a power. We also take a look at what happens to a number in scientific notation when we include a root function. So, we are going to say when we raise a value in scientific notation to a particular power, we raise the coefficient to that power. But then we multiply the exponent and that power. Here we have 3.0 × 10 - 2 and that is going to be cubed. What does this really mean? It means that our value of 3 is going to be cubed. And it also means that our power is going to multiply with that raised power. So we're gonna say 3 3 is 3x3x3 which gives us 27. And then, my exponent and my power are going to multiply with each other. So it is going to give me 10 - 6 . Now, remember, this is not the correct way to express scientific notation. The coefficient has to be a value between 1 and 10. Here, 27 is outside that range. I am going to move the decimal point over 1 to make this 2.7. And remember, if I make the coefficient smaller, that means that my exponent becomes larger. So, I moved it over by one to make it 2.7, so that means I increased this by 1. So it becomes 10 - 5 . So my answer here would be 2.7 × 10 - 5 .
We're going to say now when we take a value in scientific notation to the nth root, we raise the coefficient to the reciprocal power, and again we multiply the exponent portion by that reciprocal power value. What do I mean by the reciprocal power? Here we are taking the cube root. Cube root is the same thing as raising a number to the 1/3 power. If I took the square root of something, that's the same thing as taking that value to the 1/2 power. And if I took the 4th root of something, that's the same thing as taking it to the 1/4 power. Okay, so it's the reciprocal. So what this means is it's going to be 6 1 3 × 10 9 , also raised to the 1/3 power. So 6^(1/3) would give us 1.8 times, remember these 2 are multiplying with each other now so that's 9 times 1/3, so this becomes a 3 and this becomes a 1 to give me 3, So it becomes 10 3 . So in your calculator, you might have 2 different operations depending on what model you are using. So in your calculator, you are going to see a button that looks like this. You might have to use the second function to get to it. So what you would do is we want to do cube root here, so you hit number 3, button 3, then you look for that button with the x with the square root function there. Then you open parenthesis, plug your number in, close parenthesis. That will allow you to take the cube root of that number. Some of you may not see that button on your calculator, instead what you might see is you might see this caret button, or you might see y^x, or some of you might even see xy. So for you, what you would do is you would do parenthesis 6.0x10^9, close parenthesis. You would hit one of these three numbers here, on one of these 3, buttons here, to raise it to the power. And then you would do so hit one of those buttons. Let's say you hit this button. You would do parenthesis 1 divided by 3, close parenthesis. And then you will get your same answer as 1.8x10 to the 3rd. Make sure you go back, we are doing this together, make sure you go back and do this in your own calculator and see if you get the same exact answer as I do. You may know how to set things up but if you don't know how to plug them in correctly into your calculator, it really doesn't matter because you'll always get the wrong answer. So again, these are the operations you should apply when trying to solve a question like this. Now that we have done this, let's see if you can input these things into your calculator and get the correct answer for this example. Come back and take a look and see does your answer match up with my answer.
