1. Lesson Objectives
Be able to simplify, add, subtract, multiply, and divide square roots
Be able to simplify and evaluate algebraic expressions
Be able to simplify expressions involving exponents

2. 25 = 5 This is called the radical symbol
25 = 5
“5” is the principal square root
“25” is the radicand

3. Square roots have special properties that help you simplify, multiply, and divide them.

4. Example 2: Simplifying Square–Root Expressions
Simplify each expression.

5. Example 2: Simplifying Square–Root Expressions
Simplify each expression.

6. If a fraction has a denominator that is a square root, you can simplify it by rationalizing the denominator. To do this, multiply both the numerator and denominator by a number that produces a perfect square under the radical sign in the denominator.

7. Example 3A: Rationalizing the Denominator
Simplify by rationalizing the denominator.

8. Square roots that have the same radicand are called like radical terms
Square roots that have the same radicand are called like radical terms. Only like radical terms can be added or subtracted.

9. Example 4: Adding/Subtracting Square Roots

10. There are three different ways in which a basketball player can score points during a game. There are 1-point free throws, 2-point field goals, and 3-point field goals. An algebraic expression can represent the total points scored during a game.

11. Possible Context Clues
To translate a real-world situation into an algebraic expression, you must first determine the action being described. Then choose the operation that is indicated by the type of action and the context clues.
Action
Operation
Possible Context Clues
Combine
Add
How many total?
Combine equal groups
Multiply
How many altogether?
Separate
Subtract
How many more? How many remaining?
Separate into equal groups
Divide
How many in each group?

12. Example 1: Translating Words into Algebraic Expressions
Write an algebraic expression to represent each situation.
A. the number of apples in a basket of 12 after n more are added
B. the number of days it will take to walk 100 miles if you walk M miles per day
12 + n
Add n to 12.
Divide 100 by M.

13. Example 1: Translating Words into Algebraic Expressions
Write an algebraic expression to represent each situation.
a. Lucy’s age y years after her 18th birthday
18 + y
Add y to 18.
b. the number of seconds in h hours
3600h
Multiply h by 3600.

14. 1. Parentheses and grouping symbols. 2. Exponents.
To evaluate an algebraic expression, substitute a number for each variable and simplify by using the order of operations. One way to remember the order of operations is by using the mnemonic PEMDAS.
Order of Operations
1. Parentheses and grouping symbols.
2. Exponents.
3. Multiply and Divide from left to right.
4. Add and Subtract from left to right.

15. Example 2: Evaluating Algebraic Expressions
Evaluate the expression for the given values of the variables.
2x – xy + 4y for x = 5 and y = 2
2(5) – (5)(2) + 4(2)
Substitute 5 for x and 2 for y.
10 –
Multiply from left to right.
0 + 8
Add and subtract from left to right. 8

16. Example 2: Evaluating Algebraic Expressions
Evaluate the expression for the given values of the variables.
q2 + 4qr – r2 for r = 3 and q = 7
(7)2 + 4(7)(3) – (3)2
Substitute 3 for r and 7 for q.
49 + 4(7)(3) – 9
Evaluate exponential expressions.
– 9
Multiply from left to right.
124
Add and subtract from left to right.

17. Example 2: Evaluating Algebraic Expressions
Evaluate the expression for the given values of the variables.
x2y – xy2 + 3y for x = 2 and y = 5.
(2)2(5) – (2)(5)2 + 3(5)
Substitute 2 for x and 5 for y.
4(5) – 2(25) + 3(5)
Evaluate exponential expressions.
20 –
Multiply from left to right.
–15
Add and subtract from left to right.

18. Recall that the terms of an algebraic expression are separated by addition or subtraction symbols. Like terms have the same variables raised to the same exponents. Constant terms are like terms that always have the same value.

19. Example 3A: Simplifying Expressions
Simplify the expression.
3x2 + 2x – 3y + 4x2
3x2 + 2x – 3y + 4x2
Identify like terms.
Combine like terms. 3x2 + 4x2 = 7x2
7x2 + 2x – 3y

20. Example 3A: Simplifying Expressions
Simplify the expression.
j(6k2 + 7k) + 9jk2 – 7jk
6jk2 + 7jk + 9jk2 – 7jk
Distribute, and identify like terms.
Combine like terms. 7jk – 7jk = 0
15jk2

21. Example 3A: Simplifying Expressions
Simplify the expression.
–3(2x – xy + 3y) – 11xy.
–6x + 3xy – 9y – 11xy
Distribute, and identify like terms.
Combine like terms. 3xy – 11xy = –8xy
–6x – 8xy – 9y

22. Example 4A: Application
Apples cost $2 per pound, and grapes cost $3 per pound.
Write and simplify an expression for the total cost if you buy 10 lb of apples and grapes combined.
Let A be the number of pounds of apples.
Then 10 – A is the number of pounds of grapes.
2A + 3(10 – A)
= 2A + 30 – 3A
Distribute 3.
= 30 – A
Combine like terms.

23. In an expression of the form an, a is the base, n is the exponent, and the quantity an is called a power. The exponent indicates the number of times that the base is used as a factor.

24. When the base includes more than one symbol, it is written in parentheses.
A power includes a base and an exponent. The expression 23 is a power of 2. It is read “2 to the third power” or “2 cubed.”
Reading Math

25. Example 1A: Writing Exponential Expressions in Expanded Form
Write the expression in expanded form.
(5z)2
–s4
–s4
(5z)2
–(s  s  s  s) = –s  s  s  s
(5z)(5z)
3h3(k + 3)2
3h3(k + 3)2
3(h)(h)(h) (k + 3)(k + 3)

26. Example 1A: Writing Exponential Expressions in Expanded Form
Write the expression in expanded form.
(2a)5
3b4
3b4
(2a)5
(2a)(2a)(2a)(2a)(2a)
3  b  b  b  b
–(2x – 1)3y2
–(2x – 1)3y2
–(2x – 1)(2x – 1)(2x – 1)  y  y

28. Example 2A: Simplifying Expressions with Negative Exponents
Simplify the expression.
3–2 32
(–5)–5 5 1 æ - ç è ö ÷ ø
3  3 = 9

30. Example 3A: Using Properties of Exponents to Simplify Expressions
Simplify the expression. Assume all variables are nonzero.
3z7(–4z2)
(yz3 – 5)3 = (yz–2)3
3  (–4)  z7  z2
–12z7 + 2
y3(z–2)3
y3z(–2)(3)
–12z9

31. Example 3A: Using Properties of Exponents to Simplify Expressions
Simplify the expression. Assume all variables are nonzero.
(–2a3b)–3
(5x6)3
53(x6)3
125x(6)(3)
125x18

32. -----------------------------------------------------------
Lesson Assignment
Read Lessons*
Page 24 #22 – 40 EVEN
Page 30 #10 – 22 EVEN
Page 39 #22 – 32 EVEN
*Read the lessons? Is he kidding me? NO, I’m not. If there’s still something that’s not making sense, go online and use the book’s resource materials (videos, interactive practice, etc.)