ORCCA (Draft)Open Resources for Community College Algebra

Recall that \(\sqrt{16}=4\) because \(4^2=16\text{.}\) You can review the square root basics in Section 1.3.

For example, \(\sqrt{2}\) and \(3\sqrt{2}\) are both radical expressions. In both expressions, the number \(2\) is the radicand.

Here is an example using perfect squares to show how we can multiply square roots:

or

Whether we multiply the radicands first or take the square roots first, we get the same result. This tells us that in multiplication with radicals, we can combine factors into a single radical or separate them as needed.

Here is another example:

or

Now let's look at division. When we learned how to find the square root of a fraction, we saw that the numerators and denominators could be simplified separately. This is because of the way fractions are multiplied. We multiply the numerators and denominators independently. Here is an example of two different ways to simplify a fraction in a square root:

or

Just like in multiplication, we can separate a radical expression or combine them as needed. Let's summarize what we have learned in a theorem.

When we learned the square root properties we used perfect squares to make it easier. Now we can learn how to simplify a radical that does not contain a perfect square. Simplifying radicals is similar to simplifying fractions because we want the radicand to be as small as possible.

Let's use a calculator to compare \(\sqrt{12}\) and \(2\sqrt{3}\text{.}\)

These are equivalent expressions so let's see how we can simplify \(\sqrt{12}\) to \(2\sqrt{3}\text{.}\)

First, we will make a table of factor pairs for the number \(12\text{,}\) as we did in Section 7.3.

We can stop after the factor \(3\) because the next factor would be \(4\) which is already in our table. Now that we have the table, we want to find the largest factor that is a perfect square.

The factor pair with the largest perfect square is \(3\cdot4\text{.}\) We will use the property of multiplying radicals to separate the perfect square from the other factor. We write the perfect square first because it will end up in front of the radical.

Let's look at a few more examples.

In Theorem 8.2.3, we learned this property

which we can use to multiply square root expressions. We want to simplify each radical first to keep the radicands as small as possible. Let's look at a few examples.

Let's look at how to simplify a square root that has a negative radicand. Remember that \(\sqrt{16}=4\) because \(4^2=16\text{.}\) So what could \(\sqrt{-16}\) be equal to? There is no real number that we can square to get \(-16\text{,}\) because

To handle this situation, mathematicans separate the factor of \(\sqrt{-1}\) and represent it with the letter \(i\text{,}\) which stands for imaginary number. Imaginary numbers are widely used in electrical engineering, computer science and other fields. Now we can simplify square roots with negative numbers inside. Let's look at a few examples.

In Theorem 8.2.3, we learned this property for multiplication

but we cannot do this with addition or subtraction. Here's an example to demonstrate this:

but

We do not get the same result if we separate the radicals, so we must complete any additions and subtractions inside the radical first. Here is another example:

To add and subtract radical expressions, remember that we can only add and subtract like terms. In this case, we will call them like radicals. Compare the following:

We know the first equation is true. When we substitute \(x=\sqrt{5}\) into \(x+x=2x\text{,}\) we have the second equation. Let's look at a few more examples.

In Section 6.3, we learned how to multiply polynomials like \(2(x+3)\) and \((x+2)(x+3)\text{.}\) All the methods we learned apply when we multiply square root expressions. We will look at a few examples done with different methods.

When simplifying radicals it is useful to keep in mind that

When simplifying square root expressions, we have seen that we need to make the radicand as small as possible. Another rule is that we do not leave any radicals, which are irrational numbers, in the denominator of a fraction. In other words, we want the denominator to be rational. This process is called rationalizing the denominator.

Let's see how we can remove the square root symbol from the denominator in \(\frac{1}{\sqrt{5}}\text{.}\) If we multiply the radical by itself, we will have a perfect square that we can simplify:

To make an equivalent fraction, though, we must multiply both the numerator and denominator by the same number. If we multiply the numerator and denominator by \(\sqrt{5}\text{,}\) we have:

We can use a calculator to verify that \(\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}=0.4472\ldots\text{.}\) Let's look at a few more examples.