Roots and Powers of Algebraic Expressions

Simplifying Square Roots of Powers in Radical Expressions

Simplifying radical expressions that contain powers can be tricky. There are a few simple rules that will help you perform these simplifications with ease. This lesson will teach you how.Find a Partner

There are many instances where finding a partner can be a necessity. Dancing can be better with a partner. So can riding amusement park rides. You can be someone's 'partner in crime' or just say to them 'Howdy Pardner!'

Your partner is someone who will watch out for you and warn you of possible problems on the road ahead. This lesson will explain the importance of partners in simplifying radical expressions containing exponents. It really is the most important part.

What Do Those Radical Symbols Mean?

The radical symbol looks like this:

and is defined as a number that gives a specified quantity when multiplied to itself. For example, the square root of 25 is 5. 5 is a number that when multiplied to itself gives the specific number 25. This also means that the inverse of the square root is squared. When you square a number, taking its square root brings you back to the original number. The square root of 16 = 4; 4^2 = 16.< /P>

Radicals with Exponents

Since the radical symbol is the opposite of squared, we can make the following statement: the square root of x^2 = x.

.This just means that the square root of any term squared is equal to that term. So the square root of 4^2 is 4, the square root of b^2 is b, and so on. We can use this general rule to solve problems like this - Simplify: x^5 * y^2. The first step to solving this problem is to write each of the exponential terms out the long way. Then we match every term up with a partner. In these problems, the partners have to be the same term - no matching x's with y's. Now, since we know that the square root of x^2 is x, we can simplify this expression. Every term that is a squared term under the radical symbol can be simplified to a single term outside the radical symbol. So every x^2 under the radical will simplify to an x outside the radical, and every y with a partner will simplify to a single y outside of the square root symbol. If there is a term without a partner, it will stay under the radical. So, to simplify this equation there are two x^2, which translates to two x's outside the radical sign and one y^2, which becomes a y outside the radical, and then one x that needs to stay inside the square root. And lastly, since there are two x's outside the radical, we can combine them to equal x squared. And the answer to our problem is x^2y * the square root of x. Let's try another example - Simplify: The square root of a^4b^7c^3. First write everything out the long way, then find everyone a partner. Every pair inside the radical will simplify to one term outside the radical. And you get a^2b^3c * the square root of b * c.

Reducing Radicals Containing Numbers

The same basic rules apply when you are simplifying radicals that contain numbers, except it can be slightly more difficult to break down a number than a variable - Simplify: the square root of 75.

As with variables, first break apart the number, 5 *5 * 3, then find partners, 5^2 * 3. Any number with a partner can be removed from the radical to get your final answer, which is 5 * the square root of 3.

Lesson Summary

When simplifying radicals containing exponents, you first need to write the terms out, then find each term a partner. If there are not enough of the like terms to give everyone a partner, one can stay single. Then for each partnership, one of the terms gets placed on the outside of the radical. Any single terms will remain under the radical. Lastly, combine any terms outside the radical, if possible. For example, change b * b to b^2.