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Applying the Properties of Radicals

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Hi, and welcome. My name is Anthony Varella. And today, we're going to apply the properties of radicals. So we're going to review our properties of radicals, talk about some cautions when applying these properties, so errors to avoid, and then we'll use these properties to simplify expressions with radicals.

So first, let's go over the properties of radicals. And I should point out before we get into these properties that we're going to see variables n, a, and b, and these properties hold true when n is a positive integer and when a and b are both positive real numbers. So our first property is that if we have the nth root of a to the power of n, we can say that this equals a. This is talking about the inverse operation between an exponent and a radical. So the square root of a number squared equals that number. The cube root of a number cubed equals that number. That's what that property is getting at.

We also have a relationship between radicals and fractional exponents. So if we have the nth of a to the power of m, we can write this as our base a raised to a fractional exponent, with m in the numerator and n in the denominator. We have a product property of radicals which allows us to take the nth root of a product and express this as a series of radicals. So the nth root of a times the m through of b. Similarly, we have the quotient property of radicals. So we can express the nth root of a quotient a over b as the nth root of a divided by the nth root of b. And it's important to note, then, that the nth root of b cannot equal 0, because we can't divide by 0.

Now, a couple of cautions when applying these properties. The product and quotient properties do not apply to sums and differences. Seems straightforward, but it can be a bit tricky. So for example, we cannot say that the square root of 6 plus the square root of 10 equals the square root of 16. That would be applying our product or quotient properties to a sum. We can't do that. So we cannot say that the square root of x minus the square root of y equals the square root of x minus y. That's not a true statement.

Another caution is that an exponent under a radical can be written outside of the radical only if it's applied to everything under the radical. And this can be pretty tricky, as well. Take a look at the square root of x cubed. Now, I could write this as the square root of x quantity cubed, because our exponent 3 is applied to everything that's underneath our radical sign. However, if I have the square root of 2 x cubed, I cannot say that this is the square root of 2x quantity cubed, because this exponent 3 applies only to the x. It does not apply to the 2.

Our final caution has to do with the radicand, the expression underneath the radical, being negative. Sometimes this is OK, and sometimes it's not. So if the root is even, we have a non-real number. So the square root of negative 16 is not real, because think about squaring a negative number, or squaring 0. It's always going to be non-negative when you square it. So to take the square root of a negative number doesn't make sense with real numbers. So the square root of a negative number is not real. And this applies to all even roots. So square roots, fourth roots, sixth roots, et cetera.

But if we have an odd root, we do have a real number when taking the odd root of a negative number. So the cube root of negative 8 is negative 2, because negative 2 times negative 2 times negative 2 is negative 8. And this applies to odd roots. So cube roots, fifth roots, seventh roots, so on and so forth.

So let's get to applying these properties of radicals. So here's my first example. We have the cube root of x to the fourth times y squared over x times y. Now, before I do anything with my radical, I noticed that I can cancel out a factor of x and y. So I'm going to write this, then, as the cube root of x cubed times y.

Now, I notice that I have a product underneath my radical sign. So I can break this into two radicals-- the cube root of x cubed times the cube root of y. And now, the cube root of x cubed is making use of this property here. So I can just write that as x times the cube root of y. And that is our fully simplified expression.

In our next example, we have the fourth root of 64 x to the eighth over the fourth root of 4 x squared. Now, I'd like to be able to cancel out some terms here. But I notice that I have a fourth root in my numerator and a fourth root in the denominator. So I'm going to make use of our quotient property, and we're rewriting from this type of expression into this expression. So I'm going to put it all underneath a fourth root.

And now I can divide 64 by 4 and then divide x to the eighth by x squared. So it's 64 divided by 4 is 16. And then I can take away two factors of x, so I have x to the sixth, all underneath a fourth root. Well, now, I'm going to once again use my product property and break this down into the fourth root of 16 multiplied by the fourth root of x to the sixth power. Now, the fourth root of 16 equals 2. Another way to think about this is that 16 equals 2 to the fourth. So my four factors of two underneath a fourth root, then, that evaluates to just the integer 2.

How can we simplify the fourth root of x to the sixth power? Well, I'm first going to write this using a fractional exponent. So the fourth root of x to the sixth is the same as x raised to the power of 6 over 4. So the index to my radical is the denominator to the fractional exponent. And then I have a numerator of 6. Well, 6/4 can be simplified to 3/2. So really, I can rewrite, then, this back into a radical. This would be a square root, then, of x cubed. So I've simplified the fourth root of 16 into two, and I've simplified the fourth root of x the sixth power as the square root of x cubed.

Now, I could even simplify this further. So I can think about x cubed as being x times x times x. And I can break this down into the square root of x times the square root of x times the square root of x, making use of my product property again. Well, two factors of the square root of x, when they're multiplied together, that just equals factor of x. So I can rewrite, then, the square root of x cubed as x times the square root of x. So really, this then fully simplifies to 2x times the square root of x.

So let's review our lesson on applying the properties of radicals. So we have a couple of properties that relate to inverse operations with exponents and radicals and with fractional exponents, as well. With the product property and the quotient property, it's important to remember that these do not apply to sums and differences. So thanks for watching this tutorial on applying the properties of radicals. Hope to see you next time.