Simplifying radicals containing variables

Special care must be taken when simplifying radicals containing variables. We can start with perhaps the simplest of examples.

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

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

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

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

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

As you can see here, the answer is always x

Interesting or challenging examples of simplifying radicals containing variables

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

Try  to  write  the  expression  inside  the  radical as 

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

Notice that something is equal to 8y⁸

let us now conclude this lesson with the last example below

 Try  to  write  the  expression  inside  the  radical as 

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

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