3.10: Operations with Radicals Review
Learning Objectives
- Simplify Radicals
- Multiply Radicals
- Rationalize the Denominator
Simplifying Radicals
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 \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:
\begin{align*}1^2 = 1\end{align*} (1 is a perfect square)
\begin{align*}2^2 = 4\end{align*} (4 is a perfect square)
\begin{align*}3^2 = 9\end{align*} (9 is a perfect square)
\begin{align*}4^2 = 16\end{align*} (16 is a perfect square)
\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 \begin{align*}\sqrt{75}\end{align*}.
Break up the number 75 into 2 factors, one of which is a perfect square, and simplify:
\begin{align*}\sqrt{75} &= \sqrt{25 \times 3} \quad \ \ (25 \times 3 \ \text{is} \ 75, \ \text{and} \ 25 \ \text{is a perfect square})\\ &= \sqrt{25} \sqrt{3} \qquad \left (\text{because of the property} \ \sqrt{ab} = \sqrt{a} \sqrt{b}\right )\\ &= 5 \sqrt{3} \qquad \quad \ \left (\text{because} \ \sqrt{25} = 5\right )\end{align*}
You can check on your calculator that this is true: \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 \begin{align*}\sqrt{180}\end{align*}.
Break up the number 180 into 2 factors, one of which is a perfect square, and simplify:
\begin{align*}180 &= \sqrt{9 \times 20} \ \quad \ (9 \times 20 \ \text{is} \ 180, \ \text{and} \ 9 \ \text{is a perfect square})\\ &= \sqrt{9} \sqrt{20} \quad \ \ \ \left ( \text{because of the property} \ \sqrt{ab} = \sqrt{a} \sqrt{b}\right )\\ &= 3 \sqrt{20} \quad \quad \ \ \left ( \sqrt{9} = 3 \right )\end{align*}
We are looking good so far. However, \begin{align*}3 \sqrt{20}\end{align*} is not our final answer because we can keep simplifying the radical!
\begin{align*}180 &= 3 \sqrt{4 \times 5} \ \quad (4 \times 5 \ \text{is} \ 20, \ \text{and} \ 4 \ \text{is} \ another \ \text{perfect square})\\ &= 3 \sqrt{4} \sqrt{5}\\ &= 3 \times 2 \sqrt{5} \quad \ \left ( \sqrt{4} = 2 \right )\\ &= 6 \sqrt{5} \quad \quad \ \ \ ( 3 \times 2 = 6 )\end{align*}
This is our final answer because \begin{align*}\sqrt{5}\end{align*} does not simplify any more.
You can check on your calculator that this is true: \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?
\begin{align*}180 &= \sqrt{36 \times 5} \quad \ (36 \times 5 \ \text{is} \ 180, \ \text{and} \ 36 \ \text{is a perfect square})\\ &= \sqrt{36} \sqrt{5} \quad \ \ \left(\text{because of the property} \ \sqrt{ab} = \sqrt{a} \sqrt{b}\right )\\ &= 6 \sqrt{5} \quad \qquad \left ( \sqrt{36} = 6 \right )\end{align*}
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.
Reading Check:
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 \begin{align*} \sqrt{80}\end{align*}.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. Simplify the radical \begin{align*} \sqrt{98}\end{align*}.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Multiplying Radicals
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 \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: \begin{align*}\sqrt{a}\sqrt{b} & = \sqrt{ab}\end{align*}
We can use this to multiply (or combine) two radical expressions.
Example 3
Multiply the radicals \begin{align*} \sqrt{3} \sqrt{5}\end{align*}.
Multiply the two numbers under a single radical sign:
\begin{align*}\sqrt{3} \sqrt{5} & = \sqrt{3 \times 5}\\ & = \sqrt{15}\end{align*} Is the radical in your answer fully simplified?
Yes, we cannot simplify \begin{align*}\sqrt{15}\end{align*} any further. Therefore, \begin{align*}\sqrt{3}\sqrt{5} = \sqrt{15}\end{align*}
Example 4
Multiply the radicals \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:
Simplify each radical first:
\begin{align*}\sqrt 8 & = \sqrt{4 \times 2} && \sqrt{48} = \sqrt{16 \times 3}\\ & = \sqrt{4} \sqrt{2} && \qquad = \sqrt{16} \sqrt{3}\\ & = 2\sqrt{2} && \qquad = 4\sqrt{3}\end{align*} Now multiply your results by multiplying what is outside the radical and then multiplying what is inside the radical:
\begin{align*}\left ( 2\sqrt{2} \right )\left ( 4\sqrt{3} \right ) & = \left(2 \times 4 \sqrt{2 \times 3} \right )\\ & = 8\sqrt{6}\end{align*}
Solution #2:
Multiply the radicals first:
\begin{align*}\sqrt{8}\sqrt{48} & = \sqrt{8 \times 48}\\ & = \sqrt{348}\end{align*} Now simplify the radical:
\begin{align*}\sqrt{348} & = \sqrt{64 \times 6}\\ & = \sqrt{64} \sqrt{6}\\ & = 8\sqrt{6}\end{align*}
Reading Check:
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: \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: \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 \begin{align*}\frac{2}{3}\end{align*} is exactly the same number as \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 \begin{align*}\frac{1}{\sqrt{2}}\end{align*} , you can multiply both the top and the bottom by \begin{align*}\sqrt{2}.\end{align*}
\begin{align*} \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}\sqrt{2}} = \frac {\sqrt{2}}{\sqrt{2 \times 2}} = \frac{\sqrt{2}}{\sqrt{4}} = \frac{\sqrt{2}} {2}\end{align*}
- 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 \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, \begin{align*}\sqrt{3}\end{align*}.
\begin{align*}\frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4 \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{4 \sqrt{3}}{\sqrt{3 \times 3}} = \frac{4 \sqrt{3}}{\sqrt{9}} = \frac{4 \sqrt{3}}{3}\end{align*}
As you can see, your final answer has no radicals in the _________________.
Example 6
Rationalize the denominator of the fraction \begin{align*}\frac{6}{\sqrt{2}}\end{align*}
To rationalize, you multiply both the top and the bottom by the denominator, \begin{align*}\sqrt{2}\end{align*}.
\begin{align*}\frac{6}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6 \sqrt{2}}{\sqrt{2} \sqrt{2}} = \frac{6 \sqrt{2}}{\sqrt{2 \times 2}} = \frac{6 \sqrt{2}}{\sqrt{4}} = \frac{6 \sqrt{2}}{2}\end{align*}
Now, be sure to simplify your final fraction by reducing the numbers outside of the radical:
\begin{align*}\frac{6 \sqrt{2}}{2} = 3 \sqrt{2}\end{align*}
Reading Check:
1. True or false:
Rationalize the denominator means get all radicals out of the denominator of a fraction.
2. Rationalize the denominator and simplify: \begin{align*}\frac{3}{\sqrt{2}}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
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