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Applying the Properties of Radicals

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Today we're going to talk about the properties of radicals. So we'll start by reviewing the properties of radicals, some common mistakes, and then we'll do some examples. So let's talk about our properties of radicals. Our first property of radicals is the product property, which says that the nth root of a times b is equal to the nth root of a times the nth root of b. You can separate the two radicals.

The quotient property of radicals says that the nth root of a over b is equal to the nth root of a over the nth root of b. Again, you can separate the radicals. We have two other relationships that we use when we're simplifying with radicals.

And the first one is that the nth root of a to the n is just going to be equal to a-- the exponent and the index should be the same. And the second one is that if we have the nth root of a to the m, we can write that as a to the m over n. And these properties will hold when our value for n is a positive integer greater than 1, and when a and b are positive real numbers.

So let's go over some common mistakes people make when applying the properties of radicals. The first one is assuming that there is an addition property of radicals. So for example the thinking that the square root of 10 plus 3 can be written as the square root of 10 plus the square root of 3. However, that is not true. You cannot separate the two radicals when we have addition in between the terms.

The second common mistake that people often make is thinking that something like the square root of x to the third plus 4-- thinking that can be written as the square root of x plus 4 to the 3rd power. We cannot do that because the 3 exponent is only being applied to the x and not the 4.

We can however, rewrite this expression. The square root of x plus 4 to the 3rd we can say that that is equal to the square root of x plus 4 to the 3rd power. And that is because here we are raising the entire quantity x plus 4 to the 3rd power.

The other mistake that people commonly make when applying the properties of radicals is when you're taking a radical of a negative number, sometimes you will get a real solution, sometimes a non-real solution. So for example, if I have the cubed root of negative 8, we can say that that's equal to-- or we know that that's equal to negative 2.

And that's because a negative number raised to an odd integer will give you a negative answer. So similarly, we can take the radical all of a negative number, if the index is an odd number. We cannot, however, take the radical of a negative number, if the index is even.

So for example, if we have the 4th root of negative 16, this is going to give us a non-real answer, or non-real solution. And that's because a negative number raised to an even exponent will never give you a negative answer. And so similarly, we cannot take the radical of a negative number if the index is even.

So let's do some examples applying our properties of radicals. If you're feeling confident go ahead and try them on your own, and then check back and see how you did. So for my first example, I have the 5th root of x to the 10th times y to the 15th. So I'm going to start by using my product property of radicals, and separating this into two radicals, the 5th root of x to the 10th times the 5th root of y to the 15th.

And then I'm going to use my relationship with radicals and fractional exponents to write this first part as x to the 10 over 5 multiplied by the second part y to the 15 over 5. And then simplifying my exponents, I see that this is just x to the 3rd-- or, sorry, x to the 2nd times y to the 3rd for my final answer.

For my second example, I have the square root of a to the 4th over b squared. So now I'm going to start using my quotient property of radicals. And I'm going to separate this to be the square root of a to the 4th over the square root of b squared. And now I can use again my fractional property-- my relationship between radicals and fractional exponents.

And I'll write this first one as a to the 4/2. I know that if there's no number for my index, that's a 2, the square root over-- and then I have the square root of b squared, which I know is just b because the index and the exponent underneath are the same. And then finally, I can just simplify my exponent for a. 4/2 is just 2, so this simplifies to a squared over b for my final answer.

So let's go over our key points from today. Use the properties of radicals to simplify expressions and solve equations. Avoid these common mistakes. The properties of radicals apply only to factors, numbers, and variables combined by multiplication, and not terms, numbers and variables combined by addition or subtraction.

And the odd root of a negative value has a real solution. However, the even root of a negative value has a non-real solution. So I hope that these key points and examples helped you understand a little bit more but the properties of radicals. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Product Property of Radicals
- Property of Fractional Exponents
- Quotient Property of Radicals