### section 7.1 Learning Objectives

**7.1: Introduction to Radicals**

- Find the square roots of a perfect square
- Simplify a square root written with the radical symbol
- When the radicand is a perfect square
- When the radicand is not a perfect square
- When the root has a coefficient

- Approximate a square root
- Simplify other roots

## Find the square roots of a perfect square

We know how to square a number:

[latex]5^2=25[/latex] and [latex]\left(-5\right)^2=25[/latex]

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

### Example 1

Find the square roots of the following numbers:

- 36
- 81
- -49
- 0

- We want to find a number whose square is 36. [latex]6^2=36[/latex] therefore, the nonnegative square root of 36 is 6 and the negative square root of 36 is -6
- We want to find a number whose square is 81. [latex]9^2=81[/latex] therefore, the nonnegative square root of 81 is 9 and the negative square root of 81 is -9
- 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.
- We want to find a number whose square is 0. [latex]0^2=0[/latex] 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.

## Simplify a square root – when the radicand is a perfect square

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 [latex]\sqrt{a}[/latex]

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

By definition, the square root symbol, [latex]\sqrt{\hphantom{5}}[/latex] always means to find the nonnegativeroot, called the **principal root**.

[latex]\sqrt{-a}[/latex] is not defined, therefore [latex]\sqrt{a}[/latex] is defined for [latex]a\geq 0[/latex]

Let’s do an example similar tothe 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 2

Simplifythe following square roots:

- [latex]\sqrt{16}[/latex]
- [latex]\sqrt{9}[/latex]
- [latex]\sqrt{-9}[/latex]
- [latex]-\sqrt{25}[/latex]
- [latex]\sqrt{5^2}[/latex]
- [latex]\sqrt{0.01}[/latex]

Show Solution

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, [latex]\sqrt{a^2}=a[/latex]

In the video that follows, we simplifymore square roots using the fact that[latex]\sqrt{a^2}=a[/latex] 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 [latex]\sqrt{144}[/latex], or[latex]\sqrt{225}[/latex].

### The Product Rule for Square Roots

Given that a and b are nonnegative real numbers, [latex]\sqrt{a\cdot{b}}=\sqrt{a}\cdot\sqrt{b}[/latex]

In the examples that follow we will bring together these ideasto simplifysquare roots of numbers that are not obvious at first glance:

- square root of a square
- the product rule for square roots
- factoring

### Example 3

Simplify[latex] \sqrt{144}[/latex]

### Example 4

Simplify [latex]\sqrt{225}[/latex]

Show Solution

Caution! The square root of a product rule applieswhen you have multiplication ONLY under the square root. You cannot apply the rule to sums:

[latex]\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}[/latex]

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

[latex]\begin{array}{c}\sqrt{64+36}=\sqrt{100}=10\\\\\sqrt{64}+\sqrt{36}=8+6=14\\\\10\ne14\end{array}[/latex]

## Simplify a square root – when the radicand is not a perfect square

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 5

Simplify. [latex] \sqrt{63}[/latex]

Show Solution

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.”

Let us look at another example, where we will introduce a small shortcut.

### Example 6

Simplify: [latex]\sqrt{300}[/latex]

Show Solution

In the next example, notice there is already a number in front of the radical. The [latex]7[/latex] is being multiplied by [latex]\sqrt{24}[/latex]. This means that [latex]7[/latex] will be multiplied by any factors that come out of the radical as we simplify.

### Example 7

Simplify: [latex]7\sqrt{24}[/latex]

Show Solution

Shortcut This Way

In the next example, we introduce another 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.

- [latex]0^2=0[/latex]
- [latex]1^2=1[/latex]
- [latex]2^2=4[/latex]
- [latex]3^2=9[/latex]
- [latex]4^2=16[/latex]
- [latex]5^2=25[/latex]
- [latex]6^2=36[/latex]
- [latex]7^2=49[/latex]
- [latex]8^2=64[/latex]
- [latex]9^2=81[/latex]
- [latex]10^2=100[/latex]

### Example 8

Simplify. [latex] \sqrt{2000}[/latex]

Show Solution

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

## Approximate a square root

When the radicand is not a perfect square, sometimes it is beneficial to approximate the root rather than simplify it. The square roots of radicands that are not perfect squares are irrational numbers. This means they cannot be written exactly in fraction or decimal form. However, if we use what we know about perfect squares, it isn’t too difficult to determine which two consecutive whole numbers the square root would be between.

### Example 9

[latex]\sqrt{29}[/latex] is between which consecutive whole numbers?

Show Solution

A square root whose radicand is not a perfect square can be approximated using a calculator. Look for a calculator key that looks like [latex]\sqrt{x}[/latex] or simply [latex]\sqrt{\hspace{.08in}}[/latex] . Depending on the brand of the calculator, either the radicand is entered first and then the square root key is pressed, or the square root key is pressed first and then the radicand is entered.

### Example 10

Use a calculator to approximate [latex]\sqrt{29}[/latex] to two decimal places.

Show Solution

### Example 11

[latex]\sqrt{118}[/latex] is between which two consecutive whole numbers? Approximate [latex]\sqrt{118}[/latex] to two decimal places.

Show Solution

## Simplify other roots

Rubik’s Cube

While square roots are probably the most common radical, you can also find the third root, the fifth root, the 10throot, or really any other *n*th 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:

- 27
- 8
- -8
- 0

- We want to find a number whose cube is 27. [latex]3\cdot9=27[/latex] and [latex]9=3^2[/latex], so [latex]3\cdot3\cdot3=3^3=27[/latex]
- We want to find a number whose cube is 8. [latex]2\cdot2\cdot2=8[/latex] the cube root of 8 is 2.
- 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. [latex]-2\cdot{-2}\cdot{-2}=-8[/latex], 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.
- We want to find a number whose cube is 0. [latex]0\cdot0\cdot0[/latex], no matter how many times you multiply [latex]0[/latex] by itself, you will always get[latex]0[/latex].

The cube root of a number is written with a small number 3, called the **index**, just outside and above the radical symbol. For example, the cube root of *x*looks like [latex] \sqrt[3]{{x}}[/latex]. 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 [latex] \sqrt[3]{x}[/latex], the cube root of *x*, and [latex] 3\sqrt{x}[/latex], three *times* the *square* root of *x*. They may look similar at first, but they lead you to much different expressions!

### Example 12

Simplify each of the following:

- [latex]\sqrt[3]{27}[/latex]
- [latex]-\sqrt[3]{27}[/latex]
- [latex]\sqrt[3]{-27}[/latex]

Show Solution

Like we did with square roots, we can also use factoring to simplify cube roots such as [latex] \sqrt[3]{125}[/latex]. 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 three times (for a total of three identical factors), equals 125. Let’s factor 125 and find that number.

### Example 13

Simplify. [latex] \sqrt[3]{125}[/latex]

Show Solution

The prime factors of 125 are [latex]5\cdot5\cdot5[/latex], which can be rewritten as [latex]5^{3}[/latex]. The cube root of a cubed number is the number itself, so [latex] \sqrt[3]{{{5}^{3}}}=5[/latex]. 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.

The next example shows how to simplify a cube root when the radical is not a perfect cube.

### Example 14

Simplify: [latex]\sqrt[3]{80}[/latex]

Show Solution

Now that we have explored cube roots, we can extend these same idea to even higher order roots (those with larger indices). For example, consider the fourth root of 625, written [latex]\sqrt[4]{625}[/latex]. This is now asking us to determine “what number when multiplied by itself four times equals 625,” or equivalently, “what number raised to the fourth power is 625.” We answer this, and some variations of this, in the next example.

### Example 15

Simplify each of the following:

- [latex]\sqrt[4]{625}[/latex]
- [latex]-\sqrt[4]{625}[/latex]
- [latex]\sqrt[4]{-625}[/latex]

In each case, we are looking for a number that, when raised to the fourth power, results in the number under the radical.

- [latex]\sqrt[4]{625}[/latex]. We see that [latex]5\cdot 5\cdot 5\cdot 5=625[/latex], or [latex]5^{4}=625[/latex]. Therefore, [latex]\sqrt[4]625[/latex]=5.
- [latex]-\sqrt[4]{625}[/latex]. The minus sign in front of the radical is asking us to find the opposite of [latex]\sqrt[4]{625}[/latex]. Based on our answer above, the result would be [latex]-\sqrt{625}=-5[/latex].
- [latex]\sqrt[4]{-625}[/latex]. There is no real number that, when raised to the fourth power, will result in a negative number. Therefore, [latex]\sqrt[4]{-625}[/latex] is: Not a Real Number.

This is a good time to summarize what we have observed about negatives and radicals.

- If the index is an
**odd**number, a negative radicand will result in a**negative answer**. - If the index is an
**even**number, a negative radicand results in an answer of**Not a Real Number**. - A negative outside the radical always indicates a negative answer, regardless of whether the index is even or odd (provided the radicand is positive).

The rules we have introduced for simplifying, including the slight shortcuts, can easily be extended to a radical with any index. The next example shows a fifth root.

### Example 16

Simplify: [latex]3\sqrt[5]{64}[/latex]

Show Solution

Now that you have practiced simplifying radicals, we introduce two “Think About It” problems. In these problems, the radicals contain variables. Although you will not be held accountable for this situation, it is worth looking at to see that the same approach we have taken throughout this section applies to variables as well. You may also want to explore these for more practice with simplifying integers under radicals as well.

### think about it 1

Simplify. [latex] \sqrt[4]{32{{m}^{6}}}[/latex]

Show Solution

### think about it 2

Simplify. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]

Show Solution

To conclude the section, the following video shows more examples of simplifying cube roots (including two with variables).

## Summary

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root.

A radical expression is a mathematical way of representing the [latex]n[/latex]th root of a number. Square roots and cube roots are the most common radicals. To simplify radical expressions, look for identical factors that occur in groups of sizes matching the index.