Simplifying Radical Expressions

Learning Objective

• Simplify numeric and algebraic radical expressions.

Radical expressionsA quantity that contains a term with a radical, as in `2sqrt3` or `root(3)(8a^3bc^6)`. are expressions that include a radicalThe math symbol `sqrt`, used to denote the process of taking a root of a quantity., which is the symbol for taking a rootAny number `x` multiplied by itself a specific number of times to produce another number, such that in `x^n = y`, `x` is the `n`th root of `y` - for example, because `2^3 = 8`, `2` is the `3`rd (or cube) root of `8`. . They come in many forms, from simple and familiar, like `sqrt16`, to quite complicated, as in `3sqrt(250x^4y)`. In any case, we can use what we know about exponentsThe value that indicates the number of times another value is multiplied by itself in exponential notation. The exponent, also called the power, is written in superscript. In the term `5^3`, `5` is the base and `3` is the exponent. to make sense of such expressions.

Let’s start by exploring radicals; we’ll worry about how to simplify them later on.

A radical is a mathematical symbol used to represent the root of a number. Here’s a quick example: the phrase “the square root of `81`” is represented by the radical expression `root(2)(81)`. (In the case of square roots, this expression is commonly shortened to `sqrt81` — notice the absence of the small “`2`.”) When we find `sqrt81`, we are finding the non-negative number `r` such that `r^2=81`, which is `9`.

While square roots are probably the most common radical, we can also find the third root, the fifth root, the `10`^th root, or really any other `n`th root of a number. The `n`th root of a number can be represented by the radical expression `root(n)(x)`.

Radicals and exponents are inverse operations. For example, we know that `9^2 = 81` and `sqrt81=9`. This property can be generalized to all radicals and exponents as well: for any number, `x`, raised to an exponent `n` to produce the number `y`, the `n`th root of `y` is `x`.

We can represent this property like this: `root(n)(x^n)=x`. A warning though: it is always true if `x>=0`, and it is always true if `n` is odd. But it is not true when `x<0` and `n` is even.

Why is this the case? It is because raising any number, positive or negative, to an even power has the effect of making the new number positive. This is not the case for odd exponents. For example, think about inserting `x = -3` and `n = 2` into the formula above.

The radical would be written as `sqrt((-3)^2)`, which works out to `sqrt9`, or `3`. But our initial `x` value was `-3`, so we are left with the statement `3 = -3`. This is not true!

Taking the square root of a number requires us to break down, or factor, numbers. We need to find out what number can be multiplied by itself to produce the number we have.

If we were asked to find `sqrt16`, for example, the thought that `16 = 4^2 = 4*4` would probably come to mind. Hey now—we just factored `16` into `4*4`. 

Factoring lies at the root of simplifying radical expressions. If we understand exponents as repeated multiplication, then we can think about radicals and roots in the same way—although the way we think about the repeated multiplication under the radical sign may be a bit different than what we are used to.

Let’s explore this idea of factoring using the radical expression `root(3)(125)`. We can read this as “the third root of `125`” or “the cube root of `125`.” To simplify this expression, we are looking for a number that, when multiplied by itself two times (for a total of three identical factors), equals `125`. Let’s factor `125` and see if we can find that number.

The prime factors of `125` are `5*5*5`, which can be rewritten as `5^3`. The cube root of a cubed number is the number itself, so `root(3)(5^3)=5`. We have found the cube root, the three identical factors that equal `125`.

`125` is known as a perfect cube. This means that the cube root of `125` is an integer.

Similarly, `81`, `64`, and `49` are perfect squaresAny of the squares of the integers. Since `1^2 = 1`, `2^2 = 4`, `3^2 = 9`, etc., `1`, `4`, and `9` are perfect squares., because their square roots are also integers ( `9`, `8`, and `7`, respectively).

All even roots (square roots, fourth roots, sixth roots, etc.) are positive numbers. So, for example, `sqrt16`,  `root(4)(50)`, and `root(6)(1000)` and must be positive. This means then, that even roots exist only for positive numbers. A radical like `sqrt(-25)`  is impossible to evaluate, as no number can be multiplied by itself to produce `-25`.

Odd roots (third roots, fifth roots, seventh roots, etc.) are a different story, though. We can find an odd root of a negative number, such as `root(3)(-8)`. This radical expression simplifies to `-2` because `-2*-2*-2=-8`.  

Now let’s look at a radical that is not a perfect square root: `sqrt63`. We can find the root of this radical by using the same method as we used for `root(3)(125)`. We will factor the number under the radical (also known as the radicandThe number under the radical symbol.), `63`, looking for pairs of factors that can be expressed as a power.

So `3sqrt7`  is another way of writing `sqrt63`. We used factoring as well as the idea that `sqrt(x^2)=x` to simplify this radical. We also used another useful trick—we pulled the factors under a radical apart into separate factors, each under its own radical. This trick is called the Multiplication Property of Square Roots. It lets us pull out the perfect squares from the factors that cannot be simplified further.

Radicals will sometimes include variables as well, as in the expression `sqrt(49x^2y^4`. To simplify these radicals, we still use factoring, but we’ll have to apply the rules of exponents, too. Let’s try it:

We find that the square root of `49x^2y^4` is `7|x|y^2`. In order to check this calculation, we could square `7|x|y^2`, hoping to arrive at `49x^2y^4`. (And, in fact, you would get this expression if you evaluated `(7|x|y^2)^2`.) 

`7|x|y^2` is also known as the simplest form of this radical expression.

Simplest form is not necessarily a “better” way of representing a radical expression, it is just one way to do it. In fact, some problems may be easier to solve with a radical expression that has not been simplified. However, using simplest form can help to make sense of more complicated radical expressions.

Let’s simplify one last expression that includes variables and fractions:

Whew! That was complicated, but by taking it one step at a time, we found the solution.

A radical expression is a mathematical way of representing the `n`th root of a number. To simplify radical expressions, look for exponential terms within the radical, and then use the property `root(n)(x^n)=x` to pull out quantities. But keep in mind that while this property is always true if `x>=0`, and it is always true if `n` is odd, different things start to happen when `x<0` or `n` is even. All rules of exponents apply when simplifying radical expressions.