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

 

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:

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.

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

In the video that follows, we simplify more 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].

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

 

 

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.

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.

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.

 

 

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]

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.

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.

 

Simplify other 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.  [latex]3\cdot9=27[/latex] and [latex]9=3^2[/latex], so [latex]3\cdot3\cdot3=3^3=27[/latex]
2. We want to find a number whose cube is 8. [latex]2\cdot2\cdot2=8[/latex] 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. [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.
4. 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.

 

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.

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.

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.

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.

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.

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.