Lesson 4.02: Operations with Radicals

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1 Lesson 4.02: Operations with Radicals Take a Hike! Sheldon is planning on taking a hike through a state park. He has mapped out his route carefully. He plans to hike 3 miles to the scenic overlook, and then he will hike 2 miles to the ranger station, where his brother will pick him up. How far will Sheldon have walked? To combine these two distances, we will need to examine how to perform some operations with rational and irrational numbers. Rational and Irrational Numbers Before diving into operations, take a minute to refresh your brain on what rational and irrational numbers are. Select Next to see explanations on the different types of numbers shown in the diagram. Adding or Multiplying Rational Numbers Start with some simple rational numbers like 2 and 3. When they are added, the sum is 5, which is another rational number. When they are multiplied, the product is 6, which is another rational number. Does adding rational numbers or multiplying rational numbers always give a sum or product that is a rational number? Think of a response before reading further.

2 A number can be expressed as a ratio of two integers. That means there could be two rational numbers, a/b and c/d, as long as a, b, c, and d are all integers (and b and d cannot be 0, no dividing by 0!). So watch what happens when they are added or multiplied. The result is some integers in the numerator and some integers in the denominator. Basically a of integers, which is the definition of a rational number. Any time two rational numbers are added, the sum will be a. The same will hold true for subtraction. a/b c/d ac/bd The result is some integers multiplied together in the numerator and some integers multiplied together in the denominator. An integer times an integer equals an integer, so this is a of integers. That is the definition of a. Any time two rational numbers are multiplied, the product will be a number. Irrational Numbers Thanks to your recent experience with radicals, numbers can now be thrown into the mix. The irrational numbers used in the lesson will be numbers like 2 or 5, both of which when entered into a calculator result in non-repeating, non-terminating decimals. Adding Rational and Irrational Numbers Sheldon was walking through the state park and he was going to walk 3 miles and then 2 miles. To find his total distance, the numbers will need to be added together. However, one of his numbers is irrational. Rational Irrational Irrational 3 + = Believe it or not, that is all there is to adding a rational number to an irrational number. The terms cannot be combined. Any time a rational number is added to an irrational number, the sum will be an number. Other Rules At this point, it is common to think that a pattern has been formed. You may be thinking: An irrational number plus an irrational number equals an. An irrational number times an irrational number equals an. Unfortunately, these statements are not always true.

3 Simplifying Radicals Radicals can be used in operations just like any other number. The rules have not changed; there are just some new players in the game. However, before the operations are performed, some radicals will need to be simplified. Any number that is the square of an integer, such as 64, 9, 225, or 0, is referred to as a. Square roots work in the other direction. You are given a square and have to figure out what number times itself results in that square as the product. Square roots are indicated using the symbol, called the. The number or expression beneath this symbol is called the. Select the image of the number line to see how the square and the square root are related. While you may sometimes find perfect squares inside the square root symbol, such as 25, this is not always the case. Greatest Perfect Square Method The number line interactivity above allowed you to see the first twelve and their square roots. Finding the square roots of perfect squares is pretty straightforward, but what happens when the radicand is not a perfect square? For example, there is no integer square root of 50. While you can use your calculator to find the decimal equivalent, there is a way to simplify without using decimals. One way to simplify it is to start by finding the largest perfect square that is a factor of 50. Call this method the method. It may help to run through the perfect squares one at a time. Select each number to view the answer. So, the largest perfect square factor of 50 is 25. Rewrite 50 in factored form: Now, break 25 2 into two separate square roots. This can be done as long as you have multiplication or division inside the square root. Since 25 is a perfect square root, you know that you can rewrite 25 and 5!

4 So, 50 simplifies to. 50= An experiment can be done to prove that 50 and 5 2 represent the same value. Multiple Perfect Square Factors Sometimes an expression has multiple perfect square factors. When this happens, it is best to take the largest perfect square factor when simplifying. Check out an example of simplifying a square root with multiple perfect square factors. Keep in mind, the goal is to write the expression in simplest form. By simplifying the square root you are able to provide the exact solution. Using your calculator to write the decimal equivalent would just be providing an approximate solution since the 2 in 5 2 for example, is an irrational number, meaning the decimal continues on forever. Here are two ways the square root of 32 can be simplified. The perfect square factors of 32 are 4 and 16. The second simplification uses 16 and, as you can see, is a lot quicker and easier. The Prime Factor Method The greatest perfect square method works well when the is not too large. But what do you do when finding the greatest perfect square factor is not so easy? How would you simplify 1,260? There is another method for simplifying radicals called the method. You can use a factor tree to find the prime factors. Here's how it works. Note that it wouldn't have mattered which pair of factors you choose at each step of the tree. If all of your factors are correct, you will ultimately end up with the same prime factors, and the same simplified form of the radical expression. Check out another example of simplifying square roots using the prime factor method. Addition and Subtraction with Radicals Working with operations on radical expressions is the same process as working with variables. Only like terms can be added or subtracted =3 3 This works because both terms have 3 in them, so they can be added =

5 No simplification can happen because these are not like terms. Multiplying with Radicals Multiplication with radical expressions follows addition and subtraction in that it is similar to working with variables. There just may be some additional simplification that can occur. Notice that the get multiplied under the radical sign, while the coefficients are multiplied outside the radical sign. Look for more simplification inside the radical Distribution works the same with radicals as it has with integers and variables as well. 30

6 Review It Radical and Irrational Numbers The sums and products of two rational numbers is always rational. The sum of a rational number and an irrational number is always irrational. The product of a nonzero rational number and an irrational number is always irrational. The sums and products of two irrational numbers is either rational or irrational. Use the greatest perfect square method or prime factorization method to simplify a radical. The greatest perfect square method finds the largest perfect square in the radicand. This can then be factored out of the radicand. The prime factorization method factors the radicand into prime number factors. Pairs of factors can be pulled out of the radical. Addition and Subtraction with Radical Expressions Simplify each radical term, if possible. Identify like terms. Combine the numbers outside the like radicals and keep the radical part exactly the same. When the radicals are not the same, the coefficients outside the radicals cannot be combined. Multiplication with Radical Expressions Multiply values outside the radical. Multiply values inside the radical. Simplify where possible. Note: Remember to apply the Distribution Property when appropriate.