In this unit, we've covered the most common types of radical expressions you'll see, and how to simplify them. However, this was not an exhaustive unit; there are some kinds of radical expressions that require some different techniques for simplifying. In this final section, we'll show you some examples. 

Roots and Powers

Simplify the following expression: .

Based on what you've learned so far, it doesn't appear that this can be simplified. If you do a prime factorization of 4, you get , and since the power is less than the root, we can't simplify it. Right?

Maybe we can. Did you notice that the power (2) and the root (4) are not relatively prime (they have a common factor)? Look what happens if I write this radical in exponential form, with a fractional exponent. Remember that when converting a radical expression to an exponential expression, the power is the numerator, and the root is the denominator:

= 2

2

4

Notice that the fractional exponent can be simplified (because two and four have a common factor):

= 2

2

4

= 2

1

2

And now, we convert that back to radical format:

= 2

2

4

= 2

1

2

=

Lesson learned: if the power and the root in a radical expression have a common factor, you're not finished simplifying! Convert to exponential form, simplify the exponent, and convert back to radical form. Eventually you'll get so comfortable doing this that you'll skip the conversion process and simplify the power and root in place.

Radicals with Polynomials

In the first section of this unit, we presented the following expression: , and said that before we were done, you would be able to simplify it. Do you know how?

We begin (as we often do) by factoring:

= =  · = (x + 1)

Radicals within Radicals

If you have a radical within a radical, you may or may not be able to readily simplify the expression, but there are techniques you can try. Look at this ugly expression for example:

+

This looks ridiculously messy, but it can actually be simplified algebraically. Set the expression equal to x, and then square both sides:

x =  +
x² = ( + )²
x² = 2  + + 2()() + 2 -
x² = 4 + 2
x² = 4 + 2
x² = 6
x =

Not all messy radical expressions can be simplified using this technique, but it's worth a try if you see an expression similar to this.

Trinomial Denominator

Here's another interesting problem that might look, initially, as though it can't be simplified:

However, if you remember your Difference of Cubes factoring rule...

a³ - b³ = (a - b)(a² + ab + b²)

and we let a = , and b = 1, this rewrites as:

2 - 1 = ( - 1)( + + 1)

Notice that the second factor in the right-hand side is the same as our denominator. So if we multiply numerator and denominator by ( - 1), we'll have rationalized our denominator:

=  ·  = = - 1

Who would have thought that this ugly expression could turn out to be so simple!

1.

Simplify

2.

Simplify

3.

Simplify

4.

Simplify

5.

Simplify

6.

Simplify +

7.

Simplify