1. 2/20/2018 5:46 AM
Operations with Real Numbers and Expressions A Compare and Order any real numbers
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2. Comparing and Ordering Real Numbers
Any number that can be written as the ratio of two integers with b ≠ 0, is called a rational number.
Rational numbers can be matched with exactly one point on a number line.
There are many points on a number line for which there are no rational numbers. These numbers are called irrational numbers. Numbers such as are irrational numbers.
The rational numbers and irrational numbers make up all of the numbers on the number line and together they are called the real numbers.

3. Example 1
On the number line below indicate the approximate location of each of the real numbers.
-2, , , 2.1, ,

4. Example 1
On the number line below indicate the approximate location of each of the real numbers.
-2, , , 2.1, ,
We know , = 3.14, If we approximate the other numbers we get: /

5. Simplify Square Roots A

6. Simplify Square Roots
To simplify a square root write the radicand as a product of two factors with one of the factors being a perfect square. Look for Pairs!
Examples: Simplify the expression. 1. 2. 3.

7. Examples
Simplify each expression. 1. 2.

8. Examples
Given Find the value of x that would make the expression equivalent to
a. 9
b. 48
c. 20
d. 23

9. Greatest Common Factors and Least Common Multiples

10. Greatest Common Factor
Factors are numbers that divide evenly into a given number.
Greatest Common Factor is the largest number that divides evenly into given sets of numbers.

11. Example
Find the greatest common factor of 36x2y and 28x3y2.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 x x y
Factors of 28: 1, 2, 4, 7, 14, 28 x x x y y
For the variables look at what they have in common.
GCF is 4x2y

12. Least Common Multiple - Smallest multiple that divides evenly by every member in the given set.
Example: Find the LCM of 30xy3 and 60y2
Multiples of 30: 30, 60, 90, …
Multiples of 60: 60, 120,…
LCM of the variables: Since only one x in the problem, x must be included in LCM and LCM of y’s is the largest y exponent which is y3.
The LCM of the monomials is 60xy3.

13. Simplify/evaluate expressions involving exponents, roots, absolute value

14. Rules for Exponents Exponent Rule Examples
1. xn xm= x2 x3 = x = 27 2. 3.
x-n =

15. Rules for Exponents - continued
Exponent Rule Examples
(xm)n (x2)3 = x (4x2)4 = 44x8
(xy)n (xy)3 = x3y3 7.
8. x0 or (xy)0 = any term to the 0 power is 1.

16. Use the properties to simplify expressions. Examples:
(3a2)(4a6) 2) (4x3y5)2

17. Operations with Radicals
To add or subtract radicals, they must have the same radicand (look the same under the radical!).

18. Simplify the radical expression

19. Rationalize the denominator
Means to get the radical out of the denominator.

20. Solve radical expressions

21. Absolute Value Evaluate.
Distance from zero on the number line. So we are talking about two directions: positive/negative. Since it is distance, this is why we evaluate our answers to be positive.
Evaluate.
│-3 – 6 │= 3│7 – 20 │=
a - │b│ if a = 3 and b = -1

22. Solving Absolute Value Equations

23. Estimate to Solve Problems

24. Estimation > 5 round up < 5 round down
To calculate 10%, just move the decimal one place to the left!!!

25. Estimation
Students are going on a trip to New York City to visit the Statue of Liberty and Ellis Island. The cost for the ferry to both is $18. If the ferry holds 234 people and makes 6 trips per day, approximately how much money will the ferry make per day if every trip is full?

26. Estimation

27. Simplify expressions involving polynomials

28. Addition/Subtraction of Polynomials
Must collect like terms
If not the same variable and exact same exponent, you cannot combine them.
Subtraction - just use additive inverse!
Monomial – 1 term
Binomial – 2 terms
Trinomial – 3 terms

29. Add/Subtract the polynomials.
(3a2 – 5a + 4) + (3a – 6) =
(6x4 + 5x2 – 3x + 1) – (5x3 – x2 + 2x – 4)

30. Multiply Polynomial Expressions
Use the distributive property
Multiply every term in the first parentheses by every term in the second parentheses.
If you are multiplying two binomials, you can use FOIL to help organize the distributive property.

31. Multiply each polynomial.
(-3a2b)(5ab4) =
2. (3x + 4)(2x – 3) =
3. (2x – 5)(x2 + 3x – 10)

32. May need to use a combination of these methods too!
GCF
Difference of Squares
Ex. 21b + 24b2
= 3b(7 + 8b)
Ex. x2 – 25
= (x – 5)(x + 5)
FACTORING
Easy Trinomials
Trinomials w/leading coefficient = 1
Ex. x2 – 5x + 6
= (x – 2)(x – 3)
May need to use a combination of these methods too!

33. Simplify/Reduce rational algebraic expressions
An algebraic expression is an expression that has a polynomial in both the numerator and denominator of a fraction.