NOTES: Chapter 11. Radicals & Radical Equations. Algebra 1B COLYER Fall Student Name:
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1 NOTES: Chapter 11 Radicals & Radical Equations Algebra 1B COLYER Fall 2016 Student Name:
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3 Section 3.8 ~ Finding and Estimating Square Roots Radical: A symbol use to represent a. Radicand: The number that is the radical. Perfect Square: The number you obtain when you. In other words: Some common perfect squares: SOMETHING TO KEEP IN MIND: When you take the square root of ANY number Principal Square Root: The square root answer. Example: Negative Square Root: The of the Principal Square Root. Example: Example 1: Simplifying Square Root Expressions Simplify each expression. a. 64 b. 100 c d. 0 e. 16 f Page 3
4 Rational Numbers: Numbers that can be represented as or that or. EXAMPLES: Irrational Numbers: Numbers that CANNOT be represented as. They can be represented by that DO NOT or. EXAMPLES: Example 2: Rational and Irrational Square Roots Tell whether each expression is rational or irrational. a. 81 b c. 5 d. 4 9 e. 1 3 f Example 3: Estimating Square Roots Between what two consecutive integers is 14.52? Example 4: Approximating Square Roots With a Calculator Find to the nearest hundredth. Page 4
5 Example 5: Real-World Connection 2 2 The formula d x (2 x) gives the length d of each wire for the tower below. Find the length of the wire if x = 12 ft. Round your answer to the nearest tenth. Use the space below to complete pg 178 #9-29, 39 (you may only use a calculator for #21-24) Page 5
6 Section 3.9 ~ The Pythagorean Theorem Parts of a Right Triangle Hypotenuse: The side of a right triangle that is. Legs: The sides that. The Pythagorean Theorem: In any right triangle, the sum of the square of the lengths of the legs is equal to the square of the length of the hypotenuse. Example 1: Using the Pythagorean Theorem What is the length of the hypotenuse of the triangle below? Example 2: Real-World Connection A fire truck parks beside a building such that the base of the ladder is 16 ft from the building. The fire truck extends its ladder 30 ft as shown below. How high is the top of the ladder above the ground? Page 6
7 The Converse of the Pythagorean Theorem: If a triangle has sides of lengths a, b, and c, and with hypotenuse of length c a b c, then the triangle is a right triangle Example 3: Using the Converse of the Pythagorean Theorem Determine whether the given lengths can be sides of a right triangle. a. 5 in., 12 in., and 13 in. b. 7 m, 9 m, and 12 m Use the space below to complete pg 184 #13-17, Page 7
8 Section 11.1 ~ Simplifying Radicals Without a calculator, you can SIMPLIFY radicals to a form called. Simplest Radical Form A radical expression is in simplest radical form when all three statements are true: The radicand has no perfect square factors other than 1 The radicand has no fractions The denominator of a fraction has no radical. Things to think about Factors Commutative Property of Multiplication New Properties: Multiplication Property of Square Roots: Division Property of Square Roots: EXAMPLE 1: Removing Perfect-Square Factors a) 50 b) 18 c) EXAMPLE 2: Multiplying Two Radicals a) 3 6 b) c) Page 8
9 EXAMPLE 3: Simplifying Fractions Within Radicals a) 9 4 b) 18 4 c) Use this space to complete pg #1-5, 13-17, Page 9
10 Section 11.1 ~ Simplifying Radicals w/ Variables Definition of x 2 : Removing Variable Factors: When a variable has a nonzero, even exponent it is a. Simplifying x 2 : When a variable has an odd exponent (other than 1) it is the product of and. Simplifying x 3 : *Assume that all variables of all radicands represent nonnegative numbers. Example 1: Simplify. a) 9x 2 b) 3 45x 2 c) 45x 3 YOU TRY! Simplify each variable expression: 1) 27n 2 2) 60a 7 3) 2 x 2 y 5 Page 10
11 Practice Problems: a w 6. 6 b 10 a 7. 5 a 2 a m b 5 12 a x y a b m x y a b a b a a x y x 20 y Page 11
12 Section 11.3 ~ Solving Radical Equations Radical Equation: An equation that includes a variable in the. STEPS TO SOLVE A RADICAL EQUATION: 1. the radical on one side of the equation. 2. both sides. (*Remember the expression under the radical must be non-negative) 3. Solve the remaining. THINK: If you square a square root, what is the resulting expression? Example 1: Solving by Isolating the Radical a) x 3 = 4 Check: b) x 3 = 4 Check: c) 2 x + 7 = 5 Check: d) 2x + 5 = 7 Check: Page 12
13 Use this space to complete pg 632 #1-3, 9-12, 34-36, Page 13
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