Power and Root Functions - - Online Tutor, Practice Problems & Exam Prep

When manipulating numbers in scientific notation, raising a value to a power involves cubing the coefficient and multiplying the exponent by that power. For example,

$(3.0\times {10}^{-2}){)}^3$

becomes

$2.7\times {10}^{-5}$

after adjusting the coefficient to fall between 1 and 10 and modifying the exponent accordingly. Taking roots, such as the cube root, is akin to raising the number to the reciprocal power (e.g., the cube root is the same as raising to the 1/3 power). This results in adjusting both the coefficient and the exponent by the reciprocal power, yielding a new scientific notation expression. Calculators may have different functions for these operations, such as a root button or using a power button with the reciprocal fraction. Precision in entering these calculations is crucial to ensure the correct answer is obtained.

Powers and Root Functions

In this video we take a look at what happens when we take a number and 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're 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.

So here we have 3.0 * 10^-2 and that's going to 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, and it also means that our power is going to multiply with that raised power. So we're going to say 3³ is 3 * 3 * 3, which gives us 27. And then here my exponent and my power are going to multiply with each other, so it's 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 1.00 and 10.00. Here 27 is outside that range. I'm going to move the decimal point over one 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 increase this by 1.00 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. So what do I mean by the reciprocal power here? We're taking the cube root. OK, so there's a three here, so we're taking the cube root. Cube root is the same thing as raising a number to the one third power. If I took the square root of something, that's the same thing as taking that value to the half power. If I took the 4th root of something, that's the same thing as taking it to the one fourth power. OK, so it's the reciprocal.

So what this means is it's going to be 6 to the one third power times 10⁹ also raised to the one third power. So 6.0 to the one third would give us 1.8 times. Remember these two are multiplying with each other now, so that's 9 * 1/3. So this becomes a three and this becomes A1 to give me 3, so it becomes 10³.

So in 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 want to do cube root here, so you hit #3 button 3. Then you look for that button with the X with the square root function there. Then you'd (plug your number in). 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^ button or you might see Y to the X or some of you might even see X to the Y O for you, what you would do is you would do ( 6.0 * 10⁹ ). 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 ( 1 / 3 ) and then you'll get your same answer as 1.8 * 10³.

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 in correctly 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 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.

Powers and Root Functions

So here it says use the method discussed above determine the answer to the following question. So here we have 7.5×10-3 taken to the fifth power times the 4th root of 8.6×1021, and we're going to see what our answer is.

Now what you should do is you should do (7.5×10-3) and then you raise it to the fifth power. If you do that correctly, what you should get initially is 2.37305×10-11. Check to see you got the same answer that I did times. Remember this here is the same thing as saying 8.6×1021 in parentheses taken to the one fourth power. Remember 4th root is the same thing as one fourth power.

So if we take this to the one fourth power, what you should get at the end is 304526. 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 two sig figs. That this coefficient here has two sig figs. Or answer at the end must have two sig figs.

So what you get here is 7.2×10-6. So input that answer, input these numbers into your calculator and see if you get the same answer that I got at the end. Again, you may be able to write things down correctly on paper, but if you're struggling and punching it into your calculator then you won't ever get the correct answer. So keep practicing when it comes to inputting these values into whatever calculator model that you're using.

Here’s what students ask on this topic:

How do you raise a number in scientific notation to a power?

To raise a number in scientific notation to a power, you need to raise the coefficient to that power and multiply the exponent by the power. For example, if you have ${3.0}^{10-2}$ and you want to cube it, you would calculate ${3.0}^3$ and $10-2*\; 3$. This results in $27\; *\; 10-6$. Since 27 is not between 1 and 10, adjust it to 2.7 and increase the exponent by 1, giving you $2.7\; *\; 10-5$.

What is the process for taking the cube root of a number in scientific notation?

Taking the cube root of a number in scientific notation involves raising the coefficient to the reciprocal power (1/3) and multiplying the exponent by 1/3. For example, for $6.0\; *\; 109$, you calculate ${6.0}^{1/3}$ and $109*\; 1/3$. This results in $1.8\; *\; 103$. Ensure you use the correct calculator functions to get accurate results.

How do you adjust the coefficient in scientific notation to fall between 1 and 10?

To adjust the coefficient in scientific notation to fall between 1 and 10, move the decimal point in the coefficient and adjust the exponent accordingly. If you make the coefficient smaller, increase the exponent by the number of places you moved the decimal. For example, if you have 27 * 10^-6, move the decimal one place to get 2.7 and increase the exponent by 1, resulting in 2.7 * 10^-5.

What are the different calculator functions for raising numbers to powers and taking roots?

Calculators may have different functions for raising numbers to powers and taking roots. Some calculators have a root button (√) or a power button (x^y or y^x). To take the cube root, you might use the root button with the number 3, or use the power button with the reciprocal fraction (1/3). Ensure you input the values correctly to get accurate results.

Why is precision important when entering calculations into a calculator?

Precision is crucial when entering calculations into a calculator because incorrect input can lead to wrong answers. Understanding how to use the calculator functions properly ensures that you get accurate results, especially when dealing with scientific notation and complex operations like raising to powers or taking roots. Always double-check your inputs and follow the correct steps for your specific calculator model.