1. Unit 7 Rationals and Radicals
Rational Expressions
Reducing/Simplification
Arithmetic (multiplication and division)
Radicals
Simplifying
Arithmetic (multiplication, division, addition, subtraction)
Rationalizing denominators

2. Rational Expressions Definition:
Fractions that contain integers in their numerator and/or denominator are called _______ numbers (this is a reminder from Unit 1)
Fractions that contain ________ in their numerator and/or denominator are called rational expressions.

3. Rational Expressions
A rational expression's denominator can never be _______.
A rational expression's value is ______ when its numerator is zero, and the only way a rational expression's value can be zero is for its numerator to be ______.
When a rational expression's denominator is 1, the value of the rational expression is the value of its __________.
A rational expression's value is 1 when its numerator and denominator have the ________ (nonzero) value.
Even if a rational expression's numerator is zero, the first point applies: a rational expression's __________ can never be zero. A thing that is written ''0/0'' isn't a number. In particular, we aren't going to call it 1!
When given as a final answer, a rational expression must be reduced to lowest terms!

4. Simplifying Rational Expressions
Factor the numerator and denominator completely using the same factoring strategy we used in Unit 6
Factor out the GCF FIRST (section 5.5)
Count the number of terms
2 Terms: (section 5.7)
3 Terms: (section 5.6)
4 Terms: (section 5.5)
Cancel out factors that are common to both the numerator and denominator

5. Example 1 Original Rational Expression
Factored the numerator and denominator
Canceled out the common number factors
Looking at the a’s, there are 5 in the top and 8 in the bottom
--when canceled out, there will be 3 left in the bottom
Looking at the b’s, there are 4 in the top and 3 in the bottom
--when canceled out, there will be 1 left in the top
Final answer

6. Example 2 Original Rational Expression Factored the numerator
Factored the denominator
Canceled out the common number factors
Looking at the m’s, there are 4 in the top and 3 in the bottom—when canceled out, there will be 1 left in the bottom.
Final Answer

7. Example 3
nor does it equal

8. Example 4 Original Rational Expression
Factored the numerator using the techniques from Sect 5.6
Factored the denominator
Canceled out the common factors (2x-1)
Final Answer

9. Multiplying Rational Expressions
Our Plan of Attack
Factor all the numerators and denominators
Cancel out factors common to the numerators and denominators
Multiply the numerators
Multiply the denominators

10. Example 1 Original Multiplication Problem Final answer
Factored the numerator and denominator
Canceled out the common no. factors (3,5,7)
Looking at the a’s, there are 11 (top) & 7 (bottom)…when canceled out, there are 4 left in top. Looking at the b’s, there are 9 (top) & 10 (bottom)…when canceled out, there is 1 left in bottom.
Looking at the c’s, there are 7 (top) & 6 (bottom).. When canceled out, there is 1 left in top.
Final answer

11. Example 2 Original Multiplication Problem
Factored the trinomials using grouping (Sect. 5.6)
Factored last poly as a difference of squares (5.7)
Cancel out common factors
Canceled out the common number factors
Final Answer

12. Dividing Rational Expressions
Our Plan of Attack: Dividing rational expressions is very much like multiplying rational expressions with one extra step
KEEP – SWITCH - FLIP: Keep the first fraction, Switch to multiplication, Flip the second fraction upside down
Factor all the numerators and denominators
Cancel out factors common to the numerators and denominators
Multiply the numerators
Multiply the denominators

13. Example 1
Original Multiplication Problem Keep-Switch-Flip Factored the numerator and denominator Canceled out the common no. factors (3,5,7) Looking at the a’s, there are 6 (top) & 12 (bottom)…when canceled out, there are 6 left in bottom. Looking at the b’s, there are 7 (top) & 12 (bottom)…when canceled out, there is 5 left in bottom. Looking at the c’s, there are 8 (top) & 5 (bottom).. When canceled out, there is 3 left in top. Final answer

14. Example 2 Keep-Switch-Flip
Original Multiplication Problem
Keep-Switch-Flip
Factored the trinomials using grouping (Sect. 5.6)
Factored last poly as a difference of squares (5.7)
Cancel out common factors
Canceled out the common number factors
Final Answer

15. Radical Expressions

16. Radical Expressions
Cube Roots

17. Radical Expressions

18. Simplifying Radicals A radical is considered simplified when:
Based on what we saw in the nth root examples, we can see one of the keys in simplifying radicals is to match the index and the radicand’s exponent.
A radical is considered simplified when:
Each factor in the radicand is to a power less than the index of the radical
The radical contains NO fractions and NO negative numbers
NO radicals appear in the denominator of a fraction

19. Properties of Radicals
Product Rule for Radicals

20. Properties of Radicals
Quotient Rule for Radicals

21. Simplifying Radicals Factor the radicand
Group these factors in sets numbering the same as the index
Use the Product Rule or Quotient Rule for Radicals to rewrite the expression
Simplify (when the index and the radicands exponent match, the radical simplifies as an exponentless radicand)

22. Simplifying Radicals Example 1 Original expression
Factored the radicand into perfect squares (since index = 2)
Rewrote using the product rule
Simplified each term where
the index and radicand’s exponents matched.
Combined like terms

23. Simplifying Radicals
Original expression Factored the radicand into perfect cubes (since index =3) Rewrote using the product rule Simplified each term where the index and radicand’s exponents matched. Combined like terms

24. Arithmetic of Radicals
To Multiply Radicals
Must have same index
Multiply the coefficients
Multiply the radicands
To Divide Radicals
Divide the coefficients
Divide the radicands
To Add or Subtract Radicals
Must have same index and same radicand
Add/subtract coefficients

25. Example 1 Original Problem To Multiply Radicals Must have same index
Indexes are equal, product rule of radicals
Factored the radicand into perfect squares(since index=2)
Rewrote using the product rule
Simplified each term where the index and radicand’s exponents matched.
Combined like terms
To Multiply Radicals
Must have same index
Multiply the coefficients
Multiply the radicands

26. Example 2 To Divide Radicals Must have same index
Divide the coefficients
Divide the radicands
Original Problem
Use the quotient rule to combine the two radicals
Treat the radical as a grouping symbol and simplify the expression under the radical
Factored the radicand into perfect cubes since index= 3
Rewrote using the product rule
Simplified each term where the index and radicand’s exponents matched.
Combined like terms

27. Example 3 To Multiply Radicals Must have the same index
Multiply the coefficients
Multiply the radicands
Original Problem
Indexes are equal, product rule of radicals
Factored the radicand into perfect squares (since index=2)
Rewrote using the product rule
Simplified each term where the index and radicand’s exponents matched.
Combined like terms

28. Example 4 To Add or Subtract Radicals
Must have same index and same radicand
Add/subtract coefficients
Original Problem (it does not look like these can be combined since the radicands are not equal…but before we make a final determination, we will simplify them.)
Factored the radicand into perfect squares (since index=2)
Rewrote using the product rule
Combined like terms

29. Example 5 To Add or Subtract Radicals
Must have same index and same radicand
Add/Subtract coefficients.
Original Problem (it does not look like these can be combined since the radicands are not equal…but before we make a final determination, we will simplify them.)
Factored the radicand into perfect cubess (since index=3)
Rewrote using the product rule
Combined like terms

30. Example 6
Original Problem Distribution Property to clear grouping symbols Indexes are equal, product rule of radicals Combined like terms. Combined like terms

31. Example 7
Original Problem Distribution Property to clear grouping symbols Indexes are equal, product rule of radicals Combined like terms. Combined like terms

32. Fractional Exponents

33. Fractional Exponent Properties are the same as for whole number exponents. All the rules below apply.

34. Examples of Fractional Exponent Properties
Example 1: Example 2:

35. Examples of Fractional Exponent Properties
Example 3: Example 4:

36. Rationalizing Denominators
Simplifying Radicals: A radical is considered simplified when:
The radical contains NO fractions and NO negative numbers
NO radicals appear in the denominator of a fraction
The technique we use to get rid of any radicals in the denominator of a fraction is called rationalizing.
To rationalizing denominators, we are going to multiply the denominator by something so that the index and the radicands exponent match meaning the radical(s) in denominator will simplify as a radicand without an exponent.
Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!

37. Example 1

38. Example 2

39. Example 3

40. Example 4

41. CONJUGATES (rationalizing denominators)
The previous four examples had only ONE term in the denominator. If there are two terms, there is a slightly different technique required in order to rationalize the denominators.
We are going to multiply the denominator by its CONJUGATE (Remember … if you multiply the denominator times something, you must multiply the numerator times the exact same thing!)

42. Conjugates The conjugate of is Think: The conjugate of is Think:
Notice the only difference between an expression and its conjugate is the arithmetic in the middle.

43. Example 5

44. Example 6