Square Roots and the Order of Operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Notice the different answers in parts ⓐ and ⓑ of the example above. It is important to follow the order of operations correctly. In ⓐ, we took each square root first and then added them. In ⓑ, we added under the radical sign first and then found the square root.

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?
Consider [latex]\sqrt{9{x}^{2}}[/latex], where [latex]x\ge 0[/latex]. Can you think of an expression whose square is [latex]9{x}^{2}?[/latex]

[latex]\begin{array}{ccc}\hfill {\left(?\right)}^{2}& =& 9{x}^{2}\hfill \\ \hfill {\left(3x\right)}^{2}& =& 9{x}^{2}\text{so}\sqrt{9{x}^{2}}=3x\hfill \end{array}[/latex]
When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Operations on Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of [latex]\sqrt{2}[/latex] and [latex]3\sqrt{2}[/latex] is [latex]4\sqrt{2}[/latex]. However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression [latex]\sqrt{18}[/latex] can be written with a [latex]2[/latex] in the radicand, as [latex]3\sqrt{2}[/latex], so [latex]\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}[/latex].

Watch this video to see more examples of adding roots.

in the next video we show more examples of how to subtract radicals.

Nth Roots and Rational Exponents

Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.
Suppose we know that [latex]{a}^{3}=8[/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[/latex], we say that 2 is the cube root of 8.

The nth root of [latex]a[/latex] is a number that, when raised to the nth power, gives [latex]a[/latex]. For example, [latex]-3[/latex] is the 5th root of [latex]-243[/latex] because [latex]{\left(-3\right)}^{5}=-243[/latex]. If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex].

The principal nth root of [latex]a[/latex] is written as [latex]\sqrt[n]{a}[/latex], where [latex]n[/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[/latex] is called the index of the radical.

Simplify Radical Expressions

Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex].

To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the Product Raised to a Power Rule from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\left(ab\right)^{x}=a^{x}\cdot{b}^{x}[/latex]. So, for example, you can use the rule to rewrite [latex] {{\left( 3x \right)}^{2}}[/latex] as [latex] {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}[/latex].

Now instead of using the exponent [latex]2[/latex], use the exponent [latex] \frac{1}{2}[/latex]. The exponent is distributed in the same way.

[latex] {{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}[/latex]

And since you know that raising a number to the [latex] \frac{1}{2}[/latex] power is the same as taking the square root of that number, you can also write it this way.

[latex] \sqrt{3x}=\sqrt{3}\cdot \sqrt{x}[/latex]

Look at that—you can think of any number underneath a radical as the product of separate factors, each underneath its own radical.

This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex] \sqrt{(2\cdot 2)(2\cdot 2)(3\cdot 3})[/latex], you can rewrite the expression as the product of multiple perfect squares: [latex] \sqrt{{{2}^{2}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}[/latex].

The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.

The final answer [latex] 3\sqrt{7}[/latex] may look a bit odd, but it is in simplified form. You can read this as “three radical seven” or “three times the square root of seven.”

The following video shows more examples of how to simplify square roots that do not have perfect square radicands.

Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.

Consider the expression [latex] \sqrt{{{x}^{2}}}[/latex]. This looks like it should be equal to x, right? Test some values for x and see what happens.

In the chart below, look along each row and determine whether the value of x is the same as the value of [latex] \sqrt{{{x}^{2}}}[/latex]. Where are they equal? Where are they not equal?

After doing that for each row, look again and determine whether the value of [latex] \sqrt{{{x}^{2}}}[/latex] is the same as the value of [latex]\left|x\right|[/latex].

Notice—in cases where x is a negative number, [latex]\sqrt{x^{2}}\neq{x}[/latex]! However, in all cases [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\sqrt{x^{2}}[/latex] is always nonnegative.

We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.

Analysis of the Solution

Why did we not write [latex]b^2[/latex] as [latex]|b^2|[/latex]? Because when you square a number, you will always get a positive result, so the principal square root of [latex]\left(b^2\right)^2[/latex] will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd – including [latex]1[/latex] – add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.

In the following video, you will see more examples of how to simplify radical expressions with variables.

We will show another example where the simplified expression contains variables with both odd and even powers.

In our next example, we will start with an expression written with a rational exponent. You will see that you can use a similar process – factoring and sorting terms into squares – to simplify this expression.

Here is one more example with perfect squares.

Simplify Cube Roots

We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example, to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.

Remember that you can take the cube root of a negative expression. In the next example, we will simplify a cube root with a negative radicand.

You could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\sqrt[3]{x}[/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[/latex].

[latex] \begin{array}{l}\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\\-3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\\-27\cdot {{x}^{3}}\cdot {{y}^{3}}\cdot \sqrt[3]{{{x}^{3}}}\\-27{{x}^{3}}{{y}^{3}}\cdot x\\-27{{x}^{4}}{{y}^{3}}\end{array}[/latex]

You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.

In the following video, we show more examples of simplifying cube roots.

Simplifying Fourth Roots

Now let us move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies: find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.

An alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.

In the following video, we show another example of how to simplify a fourth and fifth root.

For our last example, we will simplify a more complicated expression, [latex]\dfrac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]. This expression has two variables, a fraction, and a radical. Let us take it step-by-step and see if using fractional exponents can help us simplify it.
We will start by simplifying the denominator since this is where the radical sign is located. Recall that an exponent in the denominator of a fraction can be rewritten as a negative exponent.

Well, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.

In our last video, we show how to use rational exponents to simplify radical expressions.

Summary

A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd and [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.

The steps to consider when simplifying a radical are outlined below.

Multiplying Radicals

You can do more than just simplify radical expressions. You can multiply and divide them, too. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:

The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that you cannot multiply a square root and a cube root using this rule. The indices of the radicals must match in order to multiply them. In our first example, we will work with integers, and then we will move on to expressions with variable radicands.

You may have also noticed that both [latex] \sqrt{18}[/latex] and [latex] \sqrt{16}[/latex] can be written as products involving perfect square factors. How would the expression change if you simplified each radical first, before multiplying? In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result.

In both cases, you arrive at the same product, [latex] 12\sqrt{2}[/latex]. It does not matter whether you multiply the radicands or simplify each radical first.

You multiply radical expressions that contain variables in the same manner. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. Look at the two examples that follow. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Note that we specify that the variable is non-negative, [latex] x\ge 0[/latex], thus allowing us to avoid the need for absolute value.

Analysis of the Solution

Even though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don’t need to write our answer with absolute value because we specified before we simplified that [latex] x\ge 0[/latex]. It is important to read the problem very well when you are doing math. Even the smallest statement like [latex] x\ge 0[/latex] can influence the way you write your answer.

In our next example, we will multiply two cube roots.

In the following video, we present more examples of how to multiply radical expressions.

This next example is slightly more complicated because there are more than two radicals being multiplied. In this case, notice how the radicals are simplified before multiplication takes place. Remember that the order you choose to use is up to you—you will find that sometimes it is easier to multiply before simplifying, and other times it is easier to simplify before multiplying. With some practice, you may be able to tell which is easier before you approach the problem, but either order will work for all problems.

In the following video, we show more examples of multiplying cube roots.