1. Chapter 5
Rational Expressions
Algebra II Notes
Mr. Heil

2. Quotients of Monomials
TSWBAT simplify fractions using the 5 laws of exponents.

3. Multiplying Fractions
Let p, q, r, and s be real numbers with q ≠ 0 and s ≠ 0, then
Example:

4. Simplifying Fractions
Let p, q, and r be real numbers with q ≠ 0 and r ≠ 0. Then
Example:

5. Laws of Exponents There are 5 Laws of Exponents.
All 5 laws start out with this generalized statement regarding the variables: Let a and b be real numbers and m and n be positive integers and a ≠ 0 and b ≠ 0 when they are divisors.

6. Laws of Exponents First Law am * an= am+n Example : x3*x5=x3+5=x8
Second Law
(ab)m = ambm
Example : (4x)3=43x3=64x3
Third Law
(am)n = am*n
Example: (x3)4=x3*4=x12

7. Laws of Exponents
Fourth Law –
Fifth Law -
Examples:

8. Zero and Negative Exponents
TSWBAT Simplify expressions using zero and negative exponents.

9. Exponents Zero Exponent – If a ≠ 0, then a0=1. Ex. 20=1 x0=1
Note: 00 is not defined.

10. Exponents
Negative Exponent – If n is a positive integer and a ≠ 0, then
Examples:

11. TSWBAT use scientific and decimal notation for numbers.
Scientific Notation
TSWBAT use scientific and decimal notation for numbers.

12. Scientific Notation
Scientific Notation – a method in which a number is expressed in the form m X 10n, where 1≤m<10, and n is an integer.
Examples –
4006 in SN is X 103.
in SN is 2.03 X 10-3.

13. Scientific Notation
Decimal Notation – The extended form of a number with all place values. This is usually the number you put into scientific notation.
Examples – 4006

14. Scientific Notation
Significant Digits – any nonzero digit or any zero that has a purpose other than placing the decimal point.
Examples – The significant digits are colored bright blue 4006,

15. Rational Algebraic Expressions
TSWBAT: Simplify rational algebraic expressions by factoring and the rules of simplifying fractions.

16. Rational Algebraic Expression
Rational Number – a number that can be expressed as a quotient of integers.
Rational Algebraic Expression – is an expression that can be expressed as a quotient of polynomials.
Example:

17. Rational Function
Rational Function – a function that is defined by a simplified rational expression in one variable.
Example -

18. Simplifying Rational Algebraic Expressions
Examples-

19. Product and Quotient of Rational Algebraic Expressions
TSWBAT multiply and divide rational algebraic expressions.

20. Products and Quotients of Rational Expressions
To find the product or quotient of two or more rational expressions we use the multiplication and division rules of fractions.
Final answers should always be expressed in simplest form. Thus the rules of simplifying fractions must be used.

21. Multiplying Fractions
Let p, q, r, and s be real numbers with q ≠ 0 and s ≠ 0, then
Example:

22. Division of Fractions
Let p, q, r, and s be real numbers with q ≠ 0, r ≠ 0, and s ≠ 0, then
Examples:

23. Sums and Differences of Rational Algebraic Expressions
TSWBAT add and subtract rational algebraic expressions.

24. Sums and Differences of Rational Expressions
If we have two or more rational expressions with the same denominator c, the following is true:
Example:

25. Sums and Differences of Rational Expressions
If the denominators are not the same, the following steps must be followed:
Step 1 – Find the LCD (Least Common Denominator) – that is the LCM of the terms in the denominator.
Step 2 - Express each fraction as an equivalent fraction with the LCD as the denominator.
Step 3 – Add or Subtract and simplify.

26. Sums and Differences of Rational Expressions
Example:
Find LCD which is 8ab2
Write equiv. fractions
Add and simplify

27. Sums and Differences of Rational Expressions
Examples

28. Complex Fractions
TSWBAT simplify complex fractions by using the rules of multiplying and dividing fractions.

29. Complex Fractions
Complex Fraction – a fraction in which the numerator, denominator, or both contain one or more fractions or powers with negative exponents.
Examples:

30. Complex Fractions To simplify these fractions there are two methods.
1. First simplify the numerator, then the denominator separately and then third divide the numerator by the denominator.

31. Complex Fractions
Example:

32. Complex Fractions
2. Multiply the numerator and denominator by the LCD of all the fractions appearing in the numerator and denominator and then combine like terms.

33. Complex Fractions
Example:

34. Fractional Coefficient Equations
TSWBAT solve equations and inequalities involving fractional coefficients.

35. Fractional Coefficients
Fractional Coefficients - To solve an equation or inequality with fractional coefficients we multiply both sides by the LCD of the fraction to turn the fraction into a normal equation or inequality.

36. Fractional Coefficients
Example:

37. TSWBAT solve and use fractional equations and word problems.

38. Fractional Equations
Fractional Equation - An equation in which a variable occurs in the denominator.
Example:

39. Fractional Equations
When solving a Fractional Equation by multiplying both sides by the LCD the new equation may not be equivalent to the original. We must check the answers at the end.

40. Fractional Equations Example Solve Find the LCD:
The LCD is: (x-2)(x-5) -> so:

41. Fractional Equations

42. Fractional Equations

43. Fractional Equations
Then we need to check each solution -> when we check x=5 we find we have a zero denominator which is not possible therefore x=5 is not a solution and only x=3 is correct.

44. Fractional Equations
Extraneous root – a root of the transformed equation that is not a root of the original equation.
Example: In the equation above 5 is an extraneous root as it solves the transformed equation but not the original.