1. Welcome to Unit 5 Our Topics for this week Radical Exponents – Review Rules for Exponents – Zero exponents – Negative Exponents Rational Expressions Simplifying Radicals Operations with Radicals

2. Laws of Exponents PRODUCT RULE OF EXPONENTS (a x ) * (a y ) = a (x + y) (KEEP THE BASE and ADD THE EXPONENTS.) EXAMPLES: Simplify: x 3 * x 2

3. Laws of Exponents QUOTIENT RULE OF EXPONENTS (a x ) / (a y ) = a (x - y) (KEEP THE BASE and SUBTRACT THE EXPONENTS) EXAMPLE: 5 7 / 5 5 = 5 7-5 = 5 2

4. Additional Laws of Exponents Anything to the zero power is 1. a 0 =1 Anything to the first power is itself. a 1 =a A negative exponent moves the term to the other side of the fraction bar. a -1 = 1/a and 1/a -1 = a

5. Eliminating Negative Exponents Move term to other side of fraction bar. a -1 = 1/a and 1/a -1 = a EXAMPLE: y 3 / y 14 = y 3 – 14 = y -11 Remove the negative sign = 1/y 11

6. Laws of Exponents POWER RULE OF EXPONENTS (a x ) y = a xy (KEEP THE BASE and MULTIPLY THE EXPONENTS.) EXAMPLE: (m 11 ) 4 = m 11*4 = m 44

7. More Examples - Power Rule When there are TWO factors (ax 3 ) 2 = a 1 * 2 * x 2 * 3 = a 2 x 6 EVALUATE: (x/y) 3 =

8. Example = = = Raise each factor to the –3 power Move all factors with negative exponents such that the exponents are positive Calculate 4 3 and 3 3, and alphabetize the variables

9. Example = = = Raise each term to the –3 power Move all terms with negative exponents such that the exponents are positive Calculate 4 3 and 3 3, and alphabetize the variables

10. Examples of Rational Exponents Rational exponents are exponents that can be expressed in the form of a fraction. x 1/2 b 2/3 c 40

11. Adding – with Rational Exponents x 1/2 + 5x 1/2 (Like Terms, add) = 6x 1/2 4y 1/2 + 6y 1/3 (Powers of y are not the same, so we cannot add. Done.)

12. Multiplying with exponents (RULE: Add exponents) (3x 1/2 )(4x 1/3 ) = 12x 1/2+1/3 Multiply coefficients = 12x 3/6+2/6 Add exponents, LCD = 12x 5/6

13. First divide coefficients, then subtract exponents = (-40/5) a 9/8 – ¼ b 2 – 1/3 c 1 – 1 Find LCD for exponents = -8 a 9/8 - 2/8 b 6/3-1/3 c 0 Leave exponents as improper fractions = -8 a 7/8 b 5/3 Dividing with exponents

14. Example: (x 1/2 ) 1/3 = x (1/2)(1/3) = x 1/6 Example: (a 4/5 b 2/3 ) 1/7 = a( 4/5 )( 1/7 )b( 2/3 )( 1/7 ) = a 4/35 b 2/21 “Power to a Power” Example

15. Simplify: (16x 3 y 4 z 3/8 ) 1/2 First, multiply exponents = 16 (1)(1/2) x (3)(1/2) y (4)(1/2) z (3/8)(1/2) = 16 1/2 x 3/2 y 4/2 z 3/16 Now convert number to radical form, reduce exponents = √16 x 3/2 y 2 z 3/16 Extract the square root = 4x 3/2 y 2 z 3/16 Example:

16. Radicals Radicals are roots. The typical radical symbol √ is considered to be a “square root” symbol. In WORD, use Insert, Symbol. √ [ 4] would be square root of 4. In TEXT, such as in the discussion, we write SQRT, for example √ [3]= SQRT[3].

17. Examples of Radicals √ [8] is “the square root of eight” The index is an understood 2 and the radicand is 8. ______ √100a 2 b is “the square root of one hundred a squared b” The index is an understood 2 and the radicand is 100a 2 b. ____ 3 √27c 6 is “the cube root of twenty-seven c to the sixth power” The index is 3 and the radicand is 27c 6. ___ 5 √-32 is “the fifth root of negative thirty-two” The index is 5 and the radicand is –32.

18. Terms with rational exponents are related to terms with radicals. Here’s how. ___ a m/n = n √a m When there is a fractional exponent, the numerator is a power, denominator is the index of the radical. Example: ___ x 2/3 = 3 √x 2

19. More Examples: ___ x 2/3 = 3 √x 2 ____ 200 4/7 = 7 √200 4 ________ _____ (36a 2 b 4 ) 1/2 = 2 √(36a 2 b 4 ) 1 = √36a 2 b 4

20. Simplifying Radicals √[25] is “the square root of twenty five” The index is an understood 2 and the radicand is 25. The simplified answer is +5 or -5 √[100] = 10 because 10*10 = 100 √[49] = 7 because 7*7 = 49

21. Simplifying Radicals Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, … A constant or variable with an EVEN exponent is also a perfect square: x 2, x 4, x 6, x 8, x 10, x 12, x 14, …

22. Simplifying Radicals Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, … A constant or variable with an exponent that is a MULTIPLE OF 3 is also a perfect cube: x 3, x 6, x 9, x 12, x 15, x 18, x 21, …

23. Example: Evaluate 16 3/2 Keep in mind that this can be rewritten like this: ___ ___ 2 √16 3, or just √16 3 (The denominator of the fractional exponents gives the root, and the numerator is a power) Because radicals and exponents are considered to be the “same level” in the order of operations, you can either deal with the radical first and the exponent second, or the exponent first and the radical second. Evaluating Fractional Exponents

24. You can either deal with the radical first and the exponent second, or the exponent first and the radical second. We will look at both ways. ROOT first: ___ √16 3 = 4 3 = 64 The square root of 16 is 4, and 4 3 is 4*4*4 = 64 RAISING to power first: ___ ____ √16 3 = √4096 = 64 16 cubed is 16*16*16 = 4096, and the square root of 4096 is 64. Evaluating Fractional Exponents

25. Example: Evaluate 4 5/2 Root first __ √4 5 = 2 5 = 32 The square root of 4 is 2, and 2 5 is 2 *2*2*2*2 = 32 RAISING to power first: ___ ____ √4 5 = √1024 = 32 4 to the fifth power is 4 *4*4*4*4 = 1024, and the square root of 1024 is 32. EXAMPLE:

26. Here are examples of when you might want to convert from radical form to rational exponent form: ___ √x 30 = x 30/2 = x 15 __ 3 √y 27 = y 27/3 = y 9 Changing Radical to Exponent Form

27. Simplify: ___ 4 √x 28 Example:

28. Simplify: ___ 4 √x 28 = x 28/4 = x 7 Example:

29. Practice Problems

30. Example – Multiply, given Fractional Exponents EVALUATE: (2x ½ )(3x ⅓) =

31. Example – DISTRIBUTE, given Fractional Exponents EVALUATE: -2x 5/6 (3x 1/2 – 4x -1/3 ) =

32. MORE PRACTICE