Simplifying Radical Expressions with Variables

When we want to simplify a radical expression with one or more variables under the radical symbol, the first thing we should look to do is factor out a square. After all, it's easier to find the $\sqrt{a^2}$ than $\sqrt{a^3}$ , and the basic rules of algebra provide us with multiple techniques to use. In this article, we'll examine the procedure for simplifying radical expressions with variables using this methodology and complete a few practice problems. Let's get started!

Simplifying radical expressions with variables by factoring out squares

Since we're going to be working with a lot of squares, we should remind ourselves that there are two square roots for every perfect square: one positive and the other negative. To use a simple example, consider the $\sqrt{9}$ . We know that $3\times 3=9$ , so we correctly identify 3 as a square root of 9. However, $-3\times \left(-3\right)=9$ as well since a negative number times a negative number yields a positive number. Therefore, -3 is a correct square root of 9 too. Both 3 and -3 are square roots of 9.

This principle applies if we aren't working with a perfect square also: there will always be positive and negative numbers or expressions when we square something. In math terminology, we can say that $a\times a=a^2$ and $-a\times \left(-a\right)=a^2$ . We generally denote such answers using a ± symbol read as "plus or minus." When we're simplifying radical expressions with variables through factoring, we almost always end up needing this symbol.

With that recap out of the way, let's start simplifying some radical expressions. Simplify $c\sqrt{a^3{c}^4}$ .

First, we need to factor the radicand as the product of a and a squared expression:

$c\sqrt{a^3{c}^4}=c\sqrt{{\left(a{c}^2\right)}^2\times a}$

Next, we can apply the Product Property of Square Roots:

$c\sqrt{{\left(a{c}^2\right)}^2}\times \sqrt{a}$

From here, we can simplify:

$c\left(\pm {ac}^2\right)\sqrt{a}$

$\pm {ac}^3\sqrt{a}$

Note how we're using the ± symbol to express that we have two answers without writing two different things. With this, we have successfully simplified the expression! Remember to work neatly on problems like this so you don't accidentally square something that shouldn't be squared or omit a radical sign somewhere. You don't want sloppy penmanship to lead to an incorrect answer!

Practice Problems

a. Simplify $\sqrt{a^3{c}^2}$

Whenever we simplify a radical expression by factoring out squares, our first move should be to rewrite the radicand in terms of squares wherever possible. In this case, that gives us:

$\sqrt{a\times a^2\times c^2}$

From here, we can simplify our answer. Remember that both a and c will have positive and negative square roots, so we need to use the ± symbol in our answer:

$\pm ac\sqrt{a}$

b. Simplify $\sqrt{a^3{b}^3{c}^3}$

This problem gives us three variables to work with, but the number of variables involved doesn't change our approach. First, we need to rewrite the radicand in terms of as many squares as possible:

$\sqrt{abc\times a^2\times b^2\times c^2}$

Now, we just need to simplify the resulting expression. All three of our variables will need the ± symbol after we take the square root, and don't forget to include the square root of abc as well:

$\sqrt{abc}\times \left(\pm abc\right)$

c. Simplify $4\sqrt{a^4{c}^4\times 9}$

We're multiplying our radical by another number this time, but what we have to do still hasn't changed. First, rewrite the radicand to include as many squares as possible. Note that we can do this with the 9 as well since it is a perfect square:

$4\sqrt{a^4{c}^4\times {\left(\pm 3\right)}^2}$

Take the square root of each side:

$4\times \left(\pm 3a^2{c}^2\right)$

Finally, distribute the 4:

$\pm 12a^2{c}^2$

It's easy to focus so intently on the variables that we lose any other terms, but we cannot let that happen. Every number is in the expression for a reason.

d. Simplify $\sqrt{\frac{a^3{b}^3}{c^3{d}^3}}$

First: rewrite the radicand using as many squares as possible. That will give us:

$\sqrt{\frac{{aba}^2{b}^2}{{cdc}^2{d}^2}}$

Next, we can apply the quotient property of square roots to make the expression easier to work with:

$\sqrt{\frac{{aba}^2{b}^2}{{cdc}^2{d}^2}}$

Then, we can take the square root of all of the terms:

$\pm \frac{ab\sqrt{ab}}{cd\sqrt{cd}}$

With so many variables to work with, our simplified expression doesn't look too simple. Still, we've simplified it substantially compared to what we started with.

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