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General Chemistry

Learn the toughest concepts covered in Chemistry with step-by-step video tutorials and practice problems by world-class tutors

5. BONUS: Mathematical Operations and Functions

Power and Root Functions -

What happens to an exponent when it is raised to a power or to an nth root?

Power and Root Functions
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Powers and Root Functions

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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 toward number in scientific notation when we include a root function. So we're going to say when we raise a value in scientific notation, toe a particular power, we raise the coefficient to that power. But then we multiply the exponents and that power. So here we have 3.0 times 10 to the negative, too, and that's gonna be cubed. So what does this really mean? Well, what that means is it means that our value of three is going to be cubed. It also means that are power is gonna multiply with that raise power. So we're gonna say three cubed is three times three times three, which gives us 27. And then here my exponents and my power are going to multiply with each other, so it's gonna give me 10 to the negative six. Now, remember, this is not the correct way to express scientific notation. The coefficient has to be a value between one and 10. Here, 27 is outside that range. I'm gonna move the decimal point over one to make this 2.7. And remember, if I make the coefficient smaller, that means that my exponents becomes larger. So I moved it over by one to make it 2.7. So that means I increase this by one. So it becomes 10 to the negative five. So my answer here would be to seven times 10 to the negative five. We're gonna say now, when we take ah value in scientific notation to the 10th root, we raised the coefficient to the reciprocal power. And again we multiply the exponents portion by that reciprocal power value. So what do I mean by the reciprocal power here? We're taking the cube root. Okay, so there's a three years, so we're taking the cube root. Cube Root is the same thing as raising a number to the one third power. If I take the square root of something, that's the same thing. Is taking that value to the half power? If I took the fourth root of something, that's the same thing as taken into the 1/4 power. Okay, so it's the reciprocal. So what? This means is, it's gonna be six to the one third power times 10 to the nine also raised the one third power. So 60 to the one third would give us 1. times. Remember, these two are multiplying with each other now, so that's nine times one third. So this becomes a three, and this becomes a one to give me three, so it becomes 10 to the three. So when your calculator you might have two different operations, depending on what model you're using. So in your calculator, you're going to see a button that looks like this. You might have to use second function to get to it. So what you would do is we wanna do que brewed here. So you hit number three bun three. Then you look for that button with the X with the square root function there, then you open parentheses. Plug your number in close parentheses. That would 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 carrot button, or you might see why to the X or some of you might even see X to the Y. So for you, what you would do is you would do parentheses 6.0 times, 10 to the nine close parentheses. You would hit one of these three numbers here on one of these three 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 parentheses, one divided by three close parentheses and then you'll get your same answer as 1.8 times 10 to the third. Make sure you go back. We're doing this together. Make sure you go back and do this in your own calculator and see if you get the same exact answers I do. You may know how to set things up, but if you don't know how to plug them incorrectly into your calculator, it really doesn't matter because you 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've done this, let's see if you can put 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
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Powers and Root Functions

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So here it says, use the method discussed above, determined the answer to the following question. So here we have 7.5 times, 10 to the negative three taken to the fifth power times the fourth root off 8.6 times. 10 to the 21 and we're gonna see what our answer is. Now what you should do is you should do parentheses. 7.5 times, 10 to negative. Three close parentheses and then you raise it to the fifth power. If you do that correctly, what you should get initially is 05 times 10 to the negative. 11. Check to see you got the same answer that I did times remember this here is the same thing as saying 86 times, 10 to the 21 in parentheses taken to the 1/4 power. Remember, Fourth Root is the same thing as 1/4 power. So if we take this to the 1/4 power, which you should get at the end is 3045 to 6. Then we multiply those two numbers together and remember here when you're multiplying or dividing the number of significant figures is the lowest, right? So this coefficient here has to sig figs that this coefficient here has to sick fix or answer at the end must have to sig figs. So what you get here 7.2 times 10 to the negative six. So input that answer and put these numbers into your calculator and see if you get the same example Answer that I got at the end again. You may be ableto write things down correctly on paper, but if you're struggling and punching it into your calculator, then you you won't ever get the correct answer. So keep practicing when it comes to in putting these values into whatever calculator model that you're using.
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