[SOUND] Let's discuss simplified radical form.

In order for an algebraic expression to be in simplified radical form, all of the

following must be true. [SOUND] The first property that must hold

is that no radicand contains a factor to a power greater than or equal to the

index of the radical. For example, the ³√y^5 would not be

considered simplified, so this is not simplified.

Because the power of the factor y, namely 5, is greater than the index of the

radical which is 3. [SOUND] The second property that must

hold is that no power of the radicand and the index of the radical have a common

factor other than 1. For example, the ninth root of x^12 would

not be simplified. Because nine and 12 have a common factor

of 3. The third property that needs to hold

[SOUND] is that no radical appears in the denominator.

For example, 2/√7 is not simplified, because we have the √7 in the

denominator. And the last property that must hold

[SOUND] is that no fraction appears within a radical.

For example, the √5/4 is not simplified, because we have this 5/4th's within the

radical. Alright, let's see an example of how we

do put an algebraic break expression into simplified radical form.

[SOUND] Let's put this expression into simplified radical form, and we're

assuming here that x and y represent positive real numbers.

The first thing we should notice here is that the radicand contains factors raised

to powers greater than the index of 4. We have the 6, as well as the 9.

Since we are simplifying a fourth root, we need to focus on the perfect fourth

power factors of the radicand, this 16x^6y^9.

Now, something has a perfect fourth power factor when its exponent is a multiple of

4. So, what do we have?

We have the ^4√16x^6y^9 is equal to the fourth root of 16, but 16 is (2)^4.

And now, we're going to extract the perfect fourth power factors here.

So, we're going to rewrite x^6 as x^4 * x^2, and we're going to rewrite y^9 as

y^8 * y. And this is equal to the ^4√2^4x^4x^2,