NOTES: Chapter 11

Radicals & Radical Equations

Algebra 1B

COLYER

Fall 2016

Student Name: _________________________________________________________________

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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

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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.

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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)

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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?

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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

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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

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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

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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

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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

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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:

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Use this space to complete pg 632 #1-3, 9-12, 34-36, 42-43