Math 10C

Real Numbers:Lesson #5

Mixed and Entire Radicals

Objective: By the end of this lesson, you will be able to:

Some radicals can be written in more than one way and still have the same value. These are called ______.

Definitions:

  • A radical in the form is called an ______.
  • A radical in the form is called a ______.

You can “pull apart” a radical (e.g. to) and get the same answer. This is called the Multiplication Property of Radicals:

We can use this property to change some entire radicals into mixed radicals. We call this ______radicals or writing a radical in ______.

Method 1: Find a Perfect Square Factor

1. Look for a factor of the radicand that is a ______.

* To make sure the radical is in simplest form, you must find the ______perfect square factor.

2.Write the radicand as a ______of the perfect square and its factor pair.

3.Break the radical apart at the sign into the product of two radicals.

4.Take the ______of the perfect square. Leave the other radical as is.

e.g. 1)Writeas a mixed radical in simplest form.

Method 2: Use Prime Factorization

1. Write the ______of the radicand.

2.Group as many prime factors as you can into ______.

3.Multiply ______number from each of these pairs together. This is the number that goes ______the radical.

4.The factors that don’t group into pairs stay in the ______. Multiply these numbers back together to get the number under the root sign.

e.g. 2)Write as a mixed radical using prime factorization.

e.g. 3)The area of the square shown below is 72 cm2. Determine the length of one side as a mixed radical in simplest form.

e.g. 4)Find the length of the missing side of the triangle. Express the answer as a mixed radical in simplest form.

We can also simplify cube roots, fourth roots, etc. by a similar process:

  • Unless you have perfect cubes, fourth powers, etc. memorized, it is probably best to use prime factorization to simplify these.
  • Instead of grouping prime factors into pairs, put them in groupsequal to the ______.

e.g. 5)Write the following radicals in simplest form:

a) b)

We can also reverse the process to convert mixed radicals to entire radicals.

e.g. 6)Write the following as entire radicals:

a) c)

Rewriting mixed radicals as whole radicals can help order them.

e.g. 7)Without using a calculator, arrange the following radicals in order from greatest to least:

, , ,

Check Your Understanding:

Is it possible to write every entire radical as a mixed radical? Explain why or why not.

Assignment:p. 218-219 #9-18, 20-22