Identify and Simplify Roots

Learning Outcomes

• Simplify principal square roots using factorization
• Use cube root notation to write cube roots
• Simplify cube roots using factorization
• Simplify square roots with variables
• Determine when a simplified root needs an absolute value
• Convert between radical and exponent notation
• Use the laws of exponents to simplify expressions with rational exponents
• Use rational exponents to simplify radical expressions

We know how to square a number:

$5^2=25$ and $\left(-5\right)^2=25$

Taking a square root is the opposite of squaring so we can make these statements:

• 5 is the nonngeative square root of 25
• -5 is the negative square root of 25

Find the square roots of the following numbers:

1. 36
2. 81
3. -49
4. 0

1. We want to find a number whose square is 36. $6^2=36$ therefore,  the nonnegative square root of 36 is 6 and the negative square root of 36 is -6
2. We want to find a number whose square is 81. $9^2=81$ therefore,  the nonnegative square root of 81 is 9 and the negative square root of 81 is -9
3. We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second. A negative number times itself is positive, and a positive number times itself is positive.  Therefore, -49 does not have square roots, there are no real number solutions to this question.
4. We want to find a number whose square is 0. $0^2=0$ therefore,  the nonnegative square root of 0 is 0.  We do not assign 0 a sign, so it has only one square root, and that is 0.

The notation that we use to express a square root for any real number, a, is as follows:

Writing a Square Root

The symbol for the square root is called a radical symbol. For a real number, a the square root of a is written as $\sqrt{a}$

The number that is written under the radical symbol is called the radicand.

By definition, the square root symbol, $\sqrt{\hphantom{5}}$ always means to find the nonnegative root, called the principal root.

$\sqrt{-a}$ is not defined, therefore $\sqrt{a}$ is defined for $a>0$

Let’s do an example similar to the example from above, this time using square root notation.  Note that using the square root notation means that you are only finding the principal root – the nonnegative root.

Example

Simplify the following square roots:

1. $\sqrt{16}$
2. $\sqrt{9}$
3. $\sqrt{-9}$
4. $\sqrt{5^2}$

The last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:

The square root of a square

For a nonnegative real number, a, $\sqrt{a^2}=a$

In the video that follows, we simplify more square roots using the fact that  $\sqrt{a^2}=a$ means finding the principal square root.

What if you are working with a number whose square you do not know right away?  We can use factoring and the product rule for square roots to find square roots such as $\sqrt{144}$, or  $\sqrt{225}$.

The Product Rule for Square Roots

Given that a and b are nonnegative real numbers, $\sqrt{a\cdot{b}}=\sqrt{a}\cdot\sqrt{b}$

In the examples that follow we will bring together these ideas to simplify square roots of numbers that are not obvious at first glance:

• square root of a square,
• the product rule for square roots
• factoring

Example

Simplify $\sqrt{144}$

Example

Simplify $\sqrt{225}$ Caution!  The square root of a product rule applies when you have multiplication ONLY under the square root. You cannot apply the rule to sums:

$\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}$

Prove this to yourself with some real numbers: let a = 64 and b = 36, then use the order of operations to simplify each expression.

$\begin{array}{c}\sqrt{64+36}=\sqrt{100}=10\\\\\sqrt{64}+\sqrt{36}=8+6=14\\\\10\ne14\end{array}$

So far, you have seen examples that are perfect squares. That is, each is a number whose square root is an integer. But many radical expressions are not perfect squares. Some of these radicals can still be simplified by finding perfect square factors. The example below illustrates how to factor the radicand, looking for pairs of factors that can be expressed as a square.

Example

Simplify. $\sqrt{63}$

The final answer $3\sqrt{7}$ 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.”

In the next example, we take a bit of a shortcut by making use of the common squares we know, instead of using prime factors. It helps to have the squares of the numbers between 0 and 10 fresh in your mind to make simplifying radicals faster.

• $0^2=0$
• $2^2=4$
• $3^2=9$
• $4^2=16$
• $5^2=25$
• $6^2=36$
• $7^2=49$
• $8^2=64$
• $9^2=81$
• $10^2=100$

Example

Simplify. $\sqrt{2,000}$

In this last video, we show examples of simplifying radicals that are not perfect squares.

Cube Roots

While square roots are probably the most common radical, you can also find the third root, the fifth root, the 10th root, or really any other nth root of a number. Just as the square root is a number that, when squared, gives the radicand, the cube root is a number that, when cubed, gives the radicand.

Find the cube roots of the following numbers:

1. 27
2. 8
3. -8
4. 0
1. We want to find a number whose cube is 27.  $3\cdot9=27$ and $9=3^2$, so $3/cdot3/cdot3=3^3=27$
2. We want to find a number whose cube is 8. $2\cdot2\cdot2=8$ the cube root of 8 is 2.
3. We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. $-2\cdot{-2}\cdot{-2}=-8$, so the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number.
4. We want to find a number whose cube is 0. $0\cdot0\cdot0$, no matter how many times you multiply $0$ by itself, you will always get $0$.

The cube root of a number is written with a small number 3, called the index, just outside and above the radical symbol. It looks like $\sqrt{{}}$. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol. Caution! Be careful to distinguish between $\sqrt{x}$, the cube root of x, and $3\sqrt{x}$, three times the square root of x. They may look similar at first, but they lead you to much different expressions!

We can also use factoring to simplify cube roots such as $\sqrt{125}$. You can read this as “the third root of 125” or “the cube root of 125.” To simplify this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. Let’s factor 125 and find that number.

Example

Simplify. $\sqrt{125}$

The prime factors of 125 are $5\cdot5\cdot5$, which can be rewritten as $5^{3}$. The cube root of a cubed number is the number itself, so $\sqrt{{{5}^{3}}}=5$. You have found the cube root, the three identical factors that when multiplied together give 125. 125 is known as a perfect cube because its cube root is an integer.

Here’s an example of how to simplify a radical that is not a perfect cube.

Example

Simplify. $\sqrt{32{{m}^{5}}}$

In the example below, we use the following idea:

$\sqrt{{{(-1)}^{3}}}=-1$

to simplify the radical.  You do not have to do this, but it may help you recognize cubes more easily when they are nonnegative.

Example

Simplify. $\sqrt{-27{{x}^{4}}{{y}^{3}}}$

In the video that follows, we show more examples if simplifying cube roots.

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

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

You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can simplify the radicals $\sqrt{-81},\ \sqrt{-64}$, and $\sqrt{-2187}$, but you cannot simplify the radicals $\sqrt[{}]{-100},\ \sqrt{-16}$, or $\sqrt{-2,500}$.

Let’s look at another example.

Example

Simplify. $\sqrt{-24{{a}^{5}}}$

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

When working with exponents and radicals:

• If n is odd, $\sqrt[n]{{{x}^{n}}}=x$.
• If n is even, $\sqrt[n]{{{x}^{n}}}=\left| x \right|$. (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)

Example

Simplify. $\sqrt{100{{x}^{2}}{{y}^{4}}}$

You can check your answer by squaring it to be sure it equals $100{{x}^{2}}{{y}^{4}}$.

In the last video, we share examples of finding cube roots with negative radicands.

Simplify Square Roots with Variables

Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as$\sqrt{16}$, to quite complicated, as in $\sqrt{250{{x}^{4}}y}$. Using factoring, you can simplify these radical expressions, too.

Simplifying Square Roots

Radical expressions will sometimes include variables as well as numbers. Consider the expression $\sqrt{9{{x}^{6}}}$. Simplifying a radical expression with variables is not as straightforward as the examples we have already shown with integers.

Consider the expression $\sqrt{{{x}^{2}}}$. This looks like it should be equal to x, right? Let’s 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 $\sqrt{{{x}^{2}}}$. Where are they equal? Where are they not equal?

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

$x$ $x^{2}$ $\sqrt{x^{2}}$ $\left|x\right|$
$−5$ 25 5 5
$−2$ 4 2 2
0 0 0 0
6 36 6 6
10 100 10 10

Notice—in cases where x is a negative number, $\sqrt{x^{2}}\neq{x}$! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases $\sqrt{x^{2}}=\left|x\right|$. You need to consider this fact when simplifying radicals that contain variables, because by definition $\sqrt{x^{2}}$ is always nonnegative.

Taking the Square Root of a Radical Expression

When finding the square root of an expression that contains variables raised to a power, consider that $\sqrt{x^{2}}=\left|x\right|$.

Examples: $\sqrt{9x^{2}}=3\left|x\right|$, and $\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|$

Let’s try it.
The goal is to find factors under the radical that are perfect squares so that you can take their square root.

Example

Simplify. $\sqrt{9{{x}^{6}}}$

Variable factors with even exponents can be written as squares. In the example above, ${{x}^{6}}={{x}^{3}}\cdot{{x}^{3}}={\left|x^3\right|}^{2}$ and

${{y}^{4}}={{y}^{2}}\cdot{{y}^{2}}={\left(|y^2\right|)}^{2}$.

Let’s try to simplify another radical expression.

Example

Simplify. $\sqrt{49{{x}^{10}}{{y}^{8}}}$

You find that the square root of $49{{x}^{10}}{{y}^{8}}$ is $7\left|{{x}^{5}}\right|{{y}^{4}}$. In order to check this calculation, you could square $7\left|{{x}^{5}}\right|{{y}^{4}}$, hoping to arrive at $49{{x}^{10}}{{y}^{8}}$. And, in fact, you would get this expression if you evaluated ${\left({7\left|{{x}^{5}}\right|{{y}^{4}}}\right)^{2}}$.

In the video that follows we show several examples of simplifying radicals with variables.

Example

Simplify. $\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}$

In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as $\sqrt{-125}$.

Rational Exponents

Roots can also be expressed as fractional exponents.  The square root of a number can be written with a radical symbol or by raising the number to the $\frac{1}{2}$ power. This is illustrated in the table below.

Exponent Form Root Form Root of a Square Simplified
${{25}^{\frac{1}{2}}}$ $\sqrt{25}$ $\sqrt{{{5}^{2}}}$ 5
${{16}^{\frac{1}{2}}}$ $\sqrt{16}$ $\sqrt{{{4}^{2}}}$ 4
${{100}^{\frac{1}{2}}}$ $\sqrt{100}$ $\sqrt{{{10}^{2}}}$ 10

Use the example below to familiarize yourself with the different ways to write square roots.

Example

Fill in the missing cells in the table.

Exponent Form Root Form Root of a Square Simplified
${{36}^{\frac{1}{2}}}$
$\sqrt{81}$
$\sqrt{{{12}^{2}}}$

In the following video, we show another example of filling in a table to connect the different notation used for roots.

We can extend the concept of writing $\sqrt{x}=x^{\frac{1}{2}}$ to cube roots. Remember, cubing a number raises it to the power of three. Notice that in these examples, the denominator of the rational exponent is the number 3.

Exponent Form

Integer

$\sqrt{8}$ ${{8}^{\tfrac{1}{3}}}$ 2
$\sqrt{8}$ ${{125}^{\tfrac{1}{3}}}$ 5
$\sqrt{1000}$ ${{1000}^{\tfrac{1}{3}}}$ 10

These examples help us model a relationship between radicals and rational exponents: namely, that the nth root of a number can be written as either $\sqrt[n]{x}$ or ${{x}^{\frac{1}{n}}}$.

Exponent Form

$\sqrt{x}$ ${{x}^{\tfrac{1}{2}}}$
$\sqrt{x}$ ${{x}^{\tfrac{1}{3}}}$
$\sqrt{x}$ ${{x}^{\tfrac{1}{4}}}$
$\sqrt[n]{x}$ ${{x}^{\tfrac{1}{n}}}$

Convert Between Radical and Exponent Notation

When faced with an expression containing a rational exponent, you can rewrite it using a radical. In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of $\frac{1}{2}$ translates to the square root, an exponent of $\frac{1}{5}$ translates to the fifth root or $\sqrt{{\hphantom{5}}}$, and $\frac{1}{8}$ translates to the eighth root or $\sqrt{{\hphantom{5}}}$ .

Example

Write $\sqrt{81}$ as an expression with a rational exponent.

Example

Express ${{(2x)}^{^{\frac{1}{3}}}}$ in radical form.

Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference—there are no parentheses! Look what happens.

Example

Express $2{{x}^{^{\frac{1}{2}}}}$ in radical form.

The next example is intended to help you practice placing a rational exponent on the appropriate terms in an expression that is written in radical form

Example

Express $4\sqrt{xy}$ with rational exponents.

In the next video, we show examples of converting between radical and exponent form.

When converting from radical to rational exponent notation, the degree of the root becomes the denominator of the exponent. If you start with a square root, you will have an exponent of $\frac{1}{2}$ on the expression in the radical (the radicand). On the other hand, if you start with an exponent of $\frac{1}{3}$ you will use a cube root. The following statement summarizes this idea.

Writing Fractional Exponents

Any radical in the form $\sqrt[n]{a}$  can be written using a fractional exponent in the form $a^{\frac{1}{n}}$.

Simplifying Radical Expressions Using Rational Exponents and the Laws of Exponents

Let’s explore some radical expressions now and see how to simplify them. Let’s start by simplifying this expression,  $\sqrt{{{a}^{6}}}$.

One method of simplifying this expression is to factor and pull out groups of $a^{3}$, as shown below in this example.

Example

Simplify. $\sqrt{{{a}^{6}}}$

You can also simplify this expression by thinking about the radical as an expression with a rational exponent, and using the principle that any radical in the form $\sqrt[n]{{{a}^{x}}}$ can be written using a fractional exponent in the form ${{a}^{\tfrac{x}{n}}}$.

Example

Simplify. $\sqrt{{{a}^{6}}}$

Note that rational exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.

Both simplification methods gave the same result, $a^{2}$. Depending on the context of the problem, it may be easier to use one method or the other, but for now, you’ll note that you were able to simplify this expression more quickly using rational exponents than when using the “pull-out” method.

Let’s try another example.

Example

Simplify. $\sqrt{81{{x}^{8}}{{y}^{3}}}$

Again, the alternative method is to work on simplifying under the radical by using factoring. For the example you just solved, it looks like this.

Example

Simplify. $\sqrt{81{{x}^{8}}{{y}^{3}}}$

The following video shows more examples of how to simplify a radical expression using rational exponents.