Upload
nguyennga
View
218
Download
2
Embed Size (px)
Citation preview
NOTES: Chapter 11
Radicals & Radical Equations
Algebra 1B
COLYER
Fall 2016
Student Name: _________________________________________________________________
Page 2
Page 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. 9
16
d. 0 e. 16 f. √64
2
Page 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. 1.44
c. 5 d. 49
e. 13 f.
√16
2
Example 3: Estimating Square Roots
Between what two consecutive integers is 14.52 ?
Example 4: Approximating Square Roots With a Calculator
Find √14.52 to the nearest hundredth.
Page 5
Example 5: Real-World Connection
The formula 2 2(2 )d x 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 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 7
The Converse of the Pythagorean Theorem:
If a triangle has sides of lengths a, b, and c, and 2 2 2a b c , then the triangle is a right triangle
with hypotenuse of length c.
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, 35-37
Page 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) 5√300
EXAMPLE 2: Multiplying Two Radicals
a) √3 ∙ √6 b) √18 ∙ √45 c) 5√30 ∙ √15
Page 9
EXAMPLE 3: Simplifying Fractions Within Radicals
a) √9
4 b) √
18
4 c) √
20
25
Use this space to complete pg 619-620 #1-5, 13-17, 28-31
Page 10
Section 11.1 ~ Simplifying Radicals w/ Variables
Definition of x2:
Removing Variable Factors:
When a variable has a nonzero, even exponent it is a ________________.
Simplifying √𝑥2:
When a variable has an odd exponent (other than 1) it is the product of ______________________
and ______________.
Simplifying √𝑥3:
*Assume that all variables of all radicands represent nonnegative numbers.
Example 1: Simplify.
a) √9𝑥2 b) 3√45𝑥2 c) −√45𝑥3
YOU TRY! Simplify each variable expression:
1) √27𝑛2 2) −√60𝑎7 3) 2√𝑥2𝑦5
Page 11
Practice Problems:
1. 64
9
2. 163 3.
121
20 4. 8a 5. 13w
6. 106ba 7. 295 aa 8. 6481m 9. 12560 ba 10. 614121 yx
11. 58 12. 21253 13. 72 14. 180 15. 3a
16. 7b 17. 9m 18. 5775 yx 19. 71127 ba 20. 4732 ba
21. 89a 22.
23. 6236 yx 24. yx2012 25. 200
745a
Page 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 13
Use this space to complete pg 632 #1-3, 9-12, 34-36, 42-43