Exercise 8 Page 623

We are asked to simplify the expression 3sqrt(12) two different ways. Let's analyze each of them one at a time.

First Way

One way to simplify the expression is to remove the greatest perfect-square factor of the denominator sqrt(12) from the radicand. In order to do this, we can use the Multiplication Property of Square Roots.

The number 12 can be factored as 3* 4. Let's apply the stated property and calculate the value of the squares, if possible.

Now that the denominator is written in the simplest form, let's simplify the whole fraction. To rationalize the denominator, we need to multiply the fraction by sqrt(3)sqrt(3). Note that the value of this fraction is 1, so it will not change the value of our expression.

We cannot simplify this fraction anymore, so sqrt(3)2 is the simplest version of the given expression.

Second Way

The other possible way to simplify the fraction is to begin by rationalizing the denominator. We can do this by multiplying the fraction by sqrt(12)sqrt(12).

Next, we need to simplify the numerator of the fraction. When analyzing the first way, we rewrote sqrt(12) as 2sqrt(3). Let's do this again!

Comparison

Both ways of simplifying gave us the same result, so we know that both of these ways are correct. Let's think about what might make one way better than the other for you!

• The first way was shorter — it was only 6 steps whereas the second way was 9 steps. If you prefer shorter solutions, use the first way.
• If you prefer to start by getting rid of any radicals in denominators, you will probably want to use the second way.