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# 3.10: Operations with Radicals Review

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Rationalize the Denominator

You learned how to simplify radicals in Algebra class, but we will review this topic to get ready for the final lessons in this chapter.

A radical is a number under a square root symbol (a square root symbol looks like this: \begin{align*}\sqrt{\;\;}\end{align*}).

• When a number is under a square root symbol \begin{align*}\sqrt{\;\;}\end{align*}, it is called a ___________________.

Remember, when we simplify a radical, we use the property ab=ab\begin{align*}\sqrt{ab} = \sqrt{a} \sqrt{b}\end{align*}. The key is to break the number inside the radical into 2 factors, one of which is a perfect square.

• Simplify a radical by splitting the number under the \begin{align*}\sqrt{\;\;}\end{align*} into _____________________.

Remember, a perfect square is the result when any number is squared.

For instance, these are examples of perfect squares:

12=1\begin{align*}1^2 = 1\end{align*} (1 is a perfect square)

22=4\begin{align*}2^2 = 4\end{align*} (4 is a perfect square)

32=9\begin{align*}3^2 = 9\end{align*} (9 is a perfect square)

42=16\begin{align*}4^2 = 16\end{align*} (16 is a perfect square)

52=25\begin{align*}5^2 = 25\end{align*} (25 is a perfect square) and so on. . .

Some more perfect squares are: 36, 49, 64, 81, 100, etc.

Example 1

Simplify the radical 75\begin{align*}\sqrt{75}\end{align*}.

Break up the number 75 into 2 factors, one of which is a perfect square, and simplify:

75=25×3  (25×3 is 75, and 25 is a perfect square)=253(because of the property ab=ab)=53 (because 25=5)

You can check on your calculator that this is true: 75=538.66\begin{align*}\sqrt{75} = 5 \sqrt{3} \approx 8.66\end{align*}

Note: In Example 1, we decided to split 75 into the factors 25 and 3. We could have picked the factors 15 and 5, since those also multiply to be 75. However, since neither 15 nor 5 is a perfect square, this choice would not have helped us simplify the radical.

Example 2

Simplify the radical 180\begin{align*}\sqrt{180}\end{align*}.

Break up the number 180 into 2 factors, one of which is a perfect square, and simplify:

180=9×20  (9×20 is 180, and 9 is a perfect square)=920   (because of the property ab=ab)=320  (9=3)

We are looking good so far. However, 320\begin{align*}3 \sqrt{20}\end{align*} is not our final answer because we can keep simplifying the radical!

180=34×5 (4×5 is 20, and 4 is another perfect square)=345=3×25 (4=2)=65   (3×2=6)

This is our final answer because 5\begin{align*}\sqrt{5}\end{align*} does not simplify any more.

You can check on your calculator that this is true: 180=6513.416\begin{align*}\sqrt{180} = 6 \sqrt{5} \approx 13.416\end{align*}

We learn from Example 2 that after you simplify, you should always check if you can simplify again. We also learn that you should try to pull out the largest perfect square possible for your first step. What if we had chosen different factors?

180=36×5 (36×5 is 180, and 36 is a perfect square)=365  (because of the property ab=ab)=65(36=6)

As you can see, we got the same answer, but in only one step! By picking the largest perfect square that goes into 180, we had a shorter problem.

• When simplifying radicals, pick the ___________________________ perfect square possible as one of your factors.

1. Fill in the blanks:

When we simplify a radical, break the number inside the radical into _________ factors, one of which is a __________________ ___________________.

2. Simplify the radical 80\begin{align*} \sqrt{80}\end{align*}.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

3. Simplify the radical 98\begin{align*} \sqrt{98}\end{align*}.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

Just like you simplified a radical by splitting it into parts, the opposite is true as well: two radicals can be multiplied together by combining them underneath a single radical.

• When you multiply two radicals, combine them under a single _____________________.

We know the property ab=ab\begin{align*}\sqrt {ab} = \sqrt{a} \sqrt{b}\end{align*}

Since this is an equation, we can switch both sides of the equal sign and the property is still true:

ab=ab

We can use this to multiply (or combine) two radical expressions.

Example 3

Multiply the radicals 35\begin{align*} \sqrt{3} \sqrt{5}\end{align*}.

Multiply the two numbers under a single radical sign:

35=3×5=15

Yes, we cannot simplify 15\begin{align*}\sqrt{15}\end{align*} any further. Therefore, 35=15\begin{align*}\sqrt{3}\sqrt{5} = \sqrt{15}\end{align*}

Example 4

Multiply the radicals 848\begin{align*} \sqrt{8}\sqrt{48}\end{align*}.

There are two different ways to do this problem. You can either:

• simplify the individual radicals first and then multiply (solution #1 below) OR
• multiply the radicals first and then simplify the final radical (solution #2 below)

Solution #1:

8=4×2=42=2248=16×3=163=43
Now multiply your results by multiplying what is outside the radical and then multiplying what is inside the radical:

(22)(43)=(2×42×3)=86

Solution #2:

848=8×48=348

348=64×6=646=86

1. Which of the two approaches to multiplying radicals in Example 4 on the previous page do you think is the simplest? Explain.

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. Multiply the radicals and simplify: 36\begin{align*}\sqrt{3} \sqrt{6}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

3. Multiply the radicals and simplify: 82\begin{align*}\sqrt{8} \sqrt{2}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

## Rationalizing the Denominator

It is always possible to express a fraction with no radicals in the denominator. This is called rationalizing the denominator.

• Rationalize the denominator means remove all __________________________ from the denominator of a fraction.

Removing radicals from the denominator is especially useful when you are adding fractions. The trick for doing this is based on the basic rule of fractions: if you multiply the top and the bottom of a fraction by the same number, the fraction is unchanged. This rule allows us to say, for instance, that 23\begin{align*}\frac{2}{3}\end{align*} is exactly the same number as 2×33×3=69\begin{align*}\frac{2 \times 3}{3 \times 3} = \frac{6}{9}\end{align*} .

To rationalize the denominator, you can multiply the top and the bottom of a fraction by whatever is in the denominator, even if it is a radical. So, in a case like 12\begin{align*}\frac{1}{\sqrt{2}}\end{align*} , you can multiply both the top and the bottom by 2.\begin{align*}\sqrt{2}.\end{align*}

12×22=222=22×2=24=22

• When you rationalize the denominator, multiply the numerator and the denominator by the ____________ that is in the denominator.

Example 5

Rationalize the denominator of the fraction 43\begin{align*}\frac{4}{\sqrt{3}}\end{align*}

The radical in the denominator is ______________.

To rationalize, you multiply both the top and the bottom by the denominator, 3\begin{align*}\sqrt{3}\end{align*}.

43×33=4333=433×3=439=433

Example 6

Rationalize the denominator of the fraction 62\begin{align*}\frac{6}{\sqrt{2}}\end{align*}

To rationalize, you multiply both the top and the bottom by the denominator, 2\begin{align*}\sqrt{2}\end{align*}.

62×22=6222=622×2=624=622

Now, be sure to simplify your final fraction by reducing the numbers outside of the radical:

622=32

1. True or false:

Rationalize the denominator means get all radicals out of the denominator of a fraction.

2. Rationalize the denominator and simplify: 32\begin{align*}\frac{3}{\sqrt{2}}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

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8 , 9 , 10

## Date Created:

Feb 23, 2012

May 12, 2014
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