Special care must be taken when simplifying radicals containing variables. We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples .

$$
\sqrt{x^2} = ??? $$

Let us look at a few examples in this form. If x = 3 or x = 5, then we have

$$
\sqrt{3^2} = \sqrt{9} = 3 = x $$

$$
\sqrt{5^2} = \sqrt{25} = 5 = x $$

These examples show that

$$
\sqrt{x^2} = x $$

Now, let us look at an example where x is a negative number. Let x = -6

$$
\sqrt{(-6)^2} = \sqrt{36} = 6 $$

When x is negative, the answer is not just x or -6 as we saw before. The answer is positive. To make sure that the answer is always positive, we need to take the absolute value.

$$ For\ any \ number \ y,\
\sqrt{y^2} = |y| $$

Now what about the cube root of x? The cube root of x will behave a little differently.

$$
\sqrt[3]{x^3} = ??? $$

If x = 2 or x = -2, the answer is not always positive.

$$
\sqrt[3]{2^3} = \sqrt[3]{8} = 2 = x $$

$$
\sqrt[3]{(-2)^3} = \sqrt[3]{-8} = -2 = x $$

As you can see here, the answer is always x

$$
For\ any \ number \ x,\ \sqrt[3]{(x)^3} = x $$

**Example #1**:

Simplify the following radical expression.

$$
\sqrt{64y^{16}} $$

The trick is to write the expression inside the radical as

$$
\sqrt{(something)^{2}} $$

Then,

$$
\sqrt{(something)^{2}} = something $$

We will need to use some properties of exponents to do this.

$$ \sqrt{64y^{16}} =
\sqrt{8^2 \times (y^{8})^2} = \sqrt{[8y^{8}]^2}$$

Notice that something is equal to 8y

$$ Therefore,
\sqrt{64y^{16}} = 8y^8 $$

Let us now conclude this lesson with the last example below

$$
\sqrt[3]{-125x^{12}y^{15}} $$

Try to write the expression inside the radical as

$$
\sqrt[3]{(something)^{3}} $$

Then,

$$
\sqrt[3]{(something)^{3}} = something $$

$$
\sqrt[3]{-125x^{12}y^{15}} = \sqrt[3]{(-5)^3(x^4)^3(y^5)^3} $$

$$
\sqrt[3]{-125x^{12}y^{15}} = \sqrt[3]{[(-5)(x^4)(y^5)]^3} $$

Therefore,

$$
\sqrt[3]{-125x^{12}y^{15}} = -5x^4y^5 $$

Take a look at the following radical expressions. We already solved them above. Do you understand how we got the answer? If so, way to go!