Notes #18: Real and Radical Expressions (Sections 10.1)

Algebra 1: Chapter 10 Notes 1

Notes #18: Real and Radical Expressions (Sections 10.1)

A. Square Roots

Complete:

02 = ____12 = ____22 = ____32 = ____42 = ____

52 = ____62 = ____72 = ____82 = ____92 = ____

102 = ____112 = ____122 = ____132 = ____142 = ____

152 = ____202 = ____302 = ____402 = ____502 = ____

The numbers you just wrote down are called ______because you ______another number to find them.

Complete:

- What number squared makes 16? ______or ______

- What number squared makes 81? ______or ______

- What number squared makes 0? ______

- What number squared makes -4? ______

You just found the ______( ) of each of these numbers. Positive numbers have ______real square roots and negative numbers have ______real square roots.

How do you know which root to list, the positive, the negative, or both?

• means the ______(or positive) square root of 25. = ____
• -means the ______square root of 25. -= ____
• means ______square roots of 25. = ____ or ____

Find the square roots of each number:

1.) 812.) 643.) 100

Simplify:

4.) 5.) 6.) 7.)

8.) Evaluate for x = 2. 9.) Evaluate for y = -1. Is

Is this a real number?this a real number?

Simplify: (if it is a polynomial, factor first!!)

10.) 11.)13.)

14.)15.) 16.)

B. Simplifying Square Roots

For problems like #16, you use one of two methods:

(1) Find perfect square factors of 18(2) Write a complete factor tree

to help you break it downfor 18 and simplify by taking out

“buddies”

Simplify each expression using BOTH methods. Then decide which you prefer:

Assume that all variables are nonnegative. (If it is a polynomial - ______first!!)

1.)2.) 3.)

4.) 5.)6.)

7.)8.) 9.)

10.) 11.)12.)

B. Multiplying Square Roots

We will be multiplying expressions like:

Steps: - (outside • outside) or
- simplify from there using your preferred method
OR
- if the numbers are already large, simplify first, then multiply, then simplify
again

*Remember, if you are multiplying polynomials, you must ______**

Multiply and simplify, if possible:

1.) 2.)3.)

4.) 5.) 6.)

7.) 8.) 9.)

10.)11.) 12.)

13.) 14.)15.)

Notes #19: Dividing Radical Expressions and the Pythagorean Theorem (Sections 10.1 and 10.2)

Section 10.1: Dividing Rational Expressions

Taking the square root of a fraction is the same as taking the square root of the ______and ______separately
OR
You can ______first and then split up the fraction

Examples:

Simplify:

1.) 2.) 3.)

4.) 5.) 6.)

7.) 8.) 9.)

However, sometimes our fractions don’t simplify as well…we end up with a radical expression in the denominator. This is NOT allowed!!

Ex: Ex:

To fix this problem:

- simplify the fraction as much as you can

- multiply this simplified fraction (both ______AND ______) by the exact

term that is still in a square root sign on the denominator

- simplify and reduce

Simplify:

10.) 11.)

12.)13.)

14.)15.)

16.)17.)

18.)19.)

20.)21.)

Section 10.2: The Pythagorean Theorem

Solve for x:

1.) 32 + 42 = x22.) 132 = 122 + x23.) x2 + 42 = 82

Right Triangles: Triangles with one ______.

hypotenuse = ______
legs = ______/

Pythagorean Theorem:

( ______)2 + (______)2 = (______)2

OR

a2 + b2 = c2

Use Pythagorean Theorem to solve for the third side of each right triangle. Leave your answer in simplified radical form:

4.)
/ 5.)

6.)
/ 7.)

8.) a = 5, b = 12, c = ? / 9.) a = 1, c = , b = ?
10.) A 15 foot ladder is leaning against a building. The bottom of the ladder is 9ft from the building. How high is the top of the ladder? / 11.) How long must a wire be to reach from the top of a 12-m telephone pole to a point on the ground 5m from the foot of the pole?

Determine whether the given lengths can be sides of a right triangle.

- use the longest length as c. Use the shorter two lengths as a and b

- plug into the Pythagorean theorem: a2 + b2 = c2

- if both sides of the equation are equal, then the triangle is ______

If both sides of the equal are not equal, then the triangle is ______

12.) 2ft, 3ft, 4ft13.) 6in, 7in, 8in 14.) 5cm, 5cm, cm

Notes #20: Other Operations on Radical Expressions (Section 10.3)

Adding and subtracting square roots is the same process as adding and subtracting ______: look for ______!!

Review: Add/Subtract

1.) 3x – 2y – 8x + 7y2.) -2mn – (-3x2) + mn – 7x2

Steps for adding/subtracting radical expressions:

• ______all radical expressions (break each term down as far as possible)
• Look for ______(underline, circle, box, etc)
• Combine like terms. Add the ______, but leave the roots ______

Add/Subtract:

1.) 2.)3.)

4.) 5.)

6.) 7.)

8.) 9.)

10.) 11.) 12.)

13.) 14.) 13.)

What was special about #14?

Steps for simplifying a fraction with a binomial in the denominator:

• Multiply the ______and ______of the fraction by the conjugate

Ex: Ex:

• Distribute in the numerator, use ______in the denominator
• Reduce only if ______terms can be simplified by the same factor

Ex: Ex:

Simplify

1.) 2.)

3.) 4.)

Notes #21: Review of Sections 10.1 – 10.3

Simplify:

1.) 2.) 3.)

4.)5.)

6.) 7.)

8.) 9.) 10.)

11.) 12.) 13.)

14.) 15.)

16.) 17.) 18.)

19.) 20.)

Notes #22

Sections 10.4: Solving Equations with radical expressions

Steps:

• Get the all alone
• Square ( )2 both sides (If you square a binomial, be sure to use ______!!)
• Solve for x
• Plug it back to check for extraneous solutions

Solve for x:

1.) 2.)

3.) 4.)

5.) 6.)

7.) 8.)

9.) 10.)

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