Simplifying Square Roots and Cube Roots
Product Property of Radicals
The Product Property of Radicals states the square root of a product is equal to the product of the square roots of the factors.
If m ≥ 0 and n ≥ 0, then √mn = √m · √n
For example, √15 = √3 · 5 = √3 · √5
In the following interactivities, you will apply the Product Property of Radicals to simplify radical expressions.
Simplifying Square Roots of Whole Numbers
When simplifying radical expressions, it is important to understand how to apply the Product Property of Radicals. In this interactivity, you will learn how apply this property to simplify square roots of whole numbers. Click the player button to begin.
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Simplifying Square Roots of Algebraic Expressions
In this interactivity, you will learn how to apply the Product Property of Radicals to simplify square roots of algebraic expressions. While solving problems of this type throughout the module, you can assume that the variables have positive values. Your understanding of factors, products, and square roots will continue to be a useful tool. Click the player button to begin.
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Simplifying Cube Roots of Integers
In this interactivity, you will continue to work with radical expressions as you simplify the cube roots of integers. You will apply the Product Property of Radicals to represent the cube root of an integer in simplest form. Click the player button to begin.
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Alternate Strategy to Simplifying Radical Expressions
During the interactivities, you learned that the first step to simplifying a radical expression is to represent the radicand as a product of its prime factors. For example,
| √45 | = | √3 · 3 · 5 |
| = | √32 · 5 | |
| = | √32 ·√5 | |
| = | 3√5 |
An alternate strategy to simplifying a radical expression is to begin by finding the largest perfect square factor of the radicand. For example,
| √45 | = | √9 · 5 |
| = | √9 · √5 | |
| = | 3√5 |
This process can be extended to simplify cube root expressions. You would begin by finding the largest perfect cube factor of the radicand. You may choose to use the strategy presented in the interactivities or the new method shown above to simplify radical expressions.
Between Which Two Consecutive Whole Numbers Does the Value Lie?
The principal square root of a non-negative whole number that is not a perfect square lies between two consecutive whole numbers. To learn more, take a look at the example below.
Example: Between which two consecutive whole numbers does √115 lie?
Notice that 115 is not a perfect square. To determine the two consecutive whole numbers that √115 lies between, begin by finding the decimal approximation.
Therefore, √115 lies between the consecutive whole numbers 10 and 11.
Simplifying Square Roots and Cube Roots Review
Now that you have learned how to simplify square roots and cube roots, it is time to test your knowledge. This interactivity will help you review the information covered throughout this topic. Click the player button to get started.
Did you answer the content review questions incorrectly? Do you want more instruction or extra practice? If so, view the videos Simplifying Square Roots of Whole Numbers, Simplifying Square Roots of Algebraic Expressions, and Simplifying Cube Roots of Whole Numbers from eMediaVASM.