1. Bellwork

2. Survey results:  Students who voted for online homework: 84%  Students who voted for paper homework: 16%  Students who wants to keep group testing: 52%  Students who prefer individual testing: 48%

3. Properties of Exponents Section 6.1 and 6.2

4. What will we learn?  You Will Learn Use zero and negative exponents.  Use the properties of exponents.  Solve real-life problems involving exponents.

5. Exponential Notation  a n = a * a * a * a…* a (where there are n factors)  The number a is the base and n is the exponent.

6. Zero and Negative Exponents  If a ≠ 0 is any real number and n is a positive integer, then  a 0 = 1  a -n = 1/a n

7. You try

9. Bellwork

10. Laws of Exponents  Product of power property  When multiplying two powers of the same base, add the exponents. a m a n = a m+n  Quotient of power property  When dividing two powers of the same base, subtract the exponents.  a m / a n = a m – n  Power of a power properties  When raising a power to a power, multiply the exponents.  (a m ) n = a mn

11. Example – Product Property  (-5) 4 * (-5) 5 =  (-5) 4+5 =  (-5) 9 =  -1953125

12. Example  x 5 * x 2 =  x 5+2 = x7x7

13. Example – Neg. Exponent  (-5) -6 (-5) 4 =  (-5) -6+4 =  (-5) -2 =

14. Example – Quotient of Powers

16. Example – Power of a Power  (2 3 ) 4 =  2 3*4 =  2 12 =  4096

17. Example  (3 4 ) 2 =  3 4*2 =  3 8 =  6561

18. Bellwork

20. Laws of Exponents  (ab) n = a n b n  When raising a product to a power, raise each factor to the power.  (a/b) n = a n / b n  When raising a quotient to a power, raise both the numerator and denominator to the power.  (a/b) -n = (b/a) n  When raising a quotient to a negative power, take the reciprocal and change the power to a positive.  a -m / b -n = b n / a m  To simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.

21. Example – Zero Exponent  (7b -3 ) 2 b 5 b =  7 2 b -3*2 b 5 b =  49 b -6+5+1 =  49b 0 =  49

22. Example – Power of Quotient

23. Basic Examples

24. Scientific Notation  Scientific Notation—shorthand way of writing very large or very small numbers.  4 x 10 13  4 and 13 zero’s  1.66 x 10 -12  0.00000000000166

25. Scientific Notation 1131,400,000,000= 1.314 x 10 11 Move the decimal behind the 1 st number How many places did you have to move the decimal? Put that number here!

26. Example – Scientific Notation 131,400,000,000 = 5,284,000 1.314 x 10 11 = 5.284 x 10 6

27. Bellwork 1. Simplify the following: 2. Simplify the following: 3. Simplify the following:

29. Finding nth Roots

30. nth root  If n is any positive integer, then the principal nth root of a is defined as:   If n is even, a and b must be positive.

31. If you assume the Power of a Power Property applies to rational exponents, then the following is true.

34. Examples:

37. Rational Exponents  For any rational exponent m/n in lowest terms, where m and n are integers and n>0, we define:   If n is even, then we require that a ≥ 0.

39. Properties of nth roots          

42. Rationalizing the Denominator  We don’t like to have radicals in the denominator, so we must rationalize to get rid of it.  Rationalizing the denominator is multiplying the top and bottom of the expression by the radical you are trying to eliminate and then simplifying.

43. More Examples