11-2 Radical Expressions Warm Up Warm Up Lesson Presentation Lesson Presentation California...
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11-2 Radical Expressions
Warm UpWarm Up
Lesson Presentation
California Standards
PreviewPreview
11-2 Radical Expressions
Warm UpIdentify the perfect square in each set.
1. 45 81 27 111 2. 156 99 8 25
3. 256 84 12 1000 4. 35 216 196 72
Write each number as a product of prime numbers.5. 36 6. 64
7. 196 8. 24
81
196
25
256
11-2 Radical Expressions
Extension of 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
California Standards
11-2 Radical Expressions
radical expressionradicand
Vocabulary
11-2 Radical ExpressionsAn expression that contains a radical sign is a radical expression. There are many types of radical expressions (such as square roots, cube roots, fourth roots, and so on), but in this chapter, you will study radical expressions that contain only square roots.
Examples of radical expressions:
The expression under a radical sign is the radicand. A radicand may contain numbers, variables, or both. It may contain one term or more than one term.
11-2 Radical Expressions
11-2 Radical ExpressionsRemember that, indicates a nonnegative square root. When you simplify a square-root expression containing variables, you must be sure your answer is not negative. For example, you might think that but this is incorrect because you do not know if x is positive or negative.
In both cases This is the correct simplification of
If x = 3, then In this case,
If x = –3, then In this case,
11-2 Radical Expressions
Additional Example 1: Simplifying Square-Root Expressions
Simplify each expression.
B.A. C.
11-2 Radical Expressions
Check It Out! Example 1
Simplify each expression.
a. b.
11-2 Radical Expressions
Check It Out! Example 1
Simplify each expression.
c. d.
11-2 Radical Expressions
11-2 Radical Expressions
Additional Example 2A: Using the Product Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Factor the radicand using perfect squares.
Product Property of Square Roots
Simplify.
11-2 Radical Expressions
Additional Example 2B: Using the Product Property of Square Roots
Simplify. All variables represent nonnegative numbers.
Product Property of Square Roots
Product Property of Square Roots
Since x is nonnegative, .
11-2 Radical Expressions
When factoring the radicand, use factors that are perfect squares. In Example 2A, you could have factored 18 as 6 3, but this contains no perfect squares.
Helpful Hint
11-2 Radical Expressions
Check It Out! Example 2a
Simplify. All variables represent nonnegative numbers.
Factor the radicand using perfect squares.
Product Property of Square Roots
Simplify.
11-2 Radical Expressions
Check It Out! Example 2b
Simplify. All variables represent nonnegative numbers.
Product Property of Square Roots
Product Property of Square Roots
Since y is nonnegative, .
11-2 Radical Expressions
Check It Out! Example 2c
Simplify. All variables represent nonnegative numbers.
Product Property of Square Roots
Factor the radicand using perfect squares.
Simplify.
11-2 Radical Expressions
11-2 Radical ExpressionsAdditional Example 3: Using the Quotient Property of
Square Roots Simplify. All variables represent nonnegative numbers.
Quotient Property of Square Roots
Simplify.
Simplify.
Quotient Property of Square Roots
Simplify.
A.B.
11-2 Radical ExpressionsCheck It Out! Example 3
Simplify. All variables represent nonnegative numbers.
Simplify.
Simplify.
Quotient Property of Square Roots
Quotient Property of Square Roots
Simplify.
a. b.
11-2 Radical ExpressionsCheck It Out! Example 3c
Simplify. All variables represent nonnegative numbers.
Quotient Property of Square Roots
Factor the radicand using perfect squares.
Simplify.
11-2 Radical Expressions
Additional Example 4A: Using the Product and Quotient Properties Together
Simplify. All variables represent nonnegative numbers.
Quotient Property
Write 108 as 36(3).
Product Property
Simplify.
11-2 Radical ExpressionsAdditional Example 4B: Using the Product and Quotient
Properties Together Simplify. All variables represent nonnegative numbers.
Quotient Property
Product Property
Simplify.
11-2 Radical Expressions
Caution!In the expression and 5 are not
common factors. is completely
simplified.
11-2 Radical ExpressionsCheck It Out! Example 4a
Simplify. All variables represent nonnegative numbers.
Quotient Property
Write 20 as 4(5).
Product Property
Simplify.
11-2 Radical ExpressionsCheck It Out! Example 4b
Simplify. All variables represent nonnegative numbers.
Quotient Property Product Property
Simplify.Write as .
11-2 Radical Expressions
Check It Out! Example 4c
Simplify. All variables represent nonnegative numbers.
Quotient Property
Simplify.
11-2 Radical ExpressionsAdditional Example 5: Application
A quadrangle on a college campus is a square with sides of 250 feet. If a student takes a shortcut by walking diagonally across the quadrangle, how far does he walk? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot.
The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.
250
250
Quadrangle
11-2 Radical ExpressionsAdditional Example 5 Continued
Solve for c.
Substitute 250 for a and b.
Simplify.
Factor 125,000 using perfect squares.
11-2 Radical Expressions
Additional Example 5 Continued
Use the Product Property of Square Roots.
Simplify.
Use a calculator and round to the nearest tenth.
The distance is ft, or about 353.6 feet.
11-2 Radical Expressions
Check It Out! Example 5
A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot.
60
60
The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.
11-2 Radical Expressions
Solve for c.
Substitute 60 for a and b.
Simplify.
Factor 7,200 using perfect squares.
Check It Out! Example 5 Continued
11-2 Radical Expressions
Use the Product Property of Square Roots.
Simplify.
Use a calculator and round to the nearest tenth.
Check It Out! Example 5 Continued
The distance is , or about 84.9 feet.
11-2 Radical ExpressionsLesson Quiz: Part I
Simplify each expression.
1.
2.
Simplify. All variables represent nonnegative numbers.
3. 4.
5. 6.
6
|x + 5|
11-2 Radical ExpressionsLesson Quiz: Part II
7. Two archaeologists leave from the same campsite. One travels 10 miles due north and the other travels 6 miles due west. How far apart are the archaeologists? Give the answer as a radical expression in simplest form. Then estimate the distance to the nearest tenth of a mile.
mi; 11.7 mi
