1. 1 1.2 Objectives ► Integer Exponents ► Rules for Working with Exponents ► Scientific Notation ► Radicals ► Rational Exponents ► Rationalizing the Denominator

2. 2 Example 1 – Exponential Notation (a) (b) (–3) 4 = (–3)  (–3)  (–3)  (–3) = 81 (c) –3 4 = –(3  3  3  3) = –81

3. 3 Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5  5  5 is written as 5 3.

4. 4 Example 2 – Zero and Negative Exponents (a) (b) (c) = 1

5. 5 Integer Exponents

6. 6 Rules for Working with Exponents Familiarity with the following rules is essential for our work with exponents and bases. In the table the bases a and b are real numbers, and the exponents m and n are integers.

7. 7

8. 8 Example 4 – Simplifying Expressions with Exponents Simplify: (2a 3 b 2 )(3ab 4 ) 3 Solution: (2a 3 b 2 )(3ab 4 ) 3 = (2a 3 b 2 )[3 3 a 3 (b 4 ) 3 ] = (2a 3 b 2 )(27a 3 b 12 ) = (2)(27)a 3 a 3 b 2 b 12 = 54a 6 b 14 Law 4: (ab) n = a n b n Group factors with the same base Law 3: (a m ) n = a mn Law 1: a m b n = a m + n

9. 9 Example 4 Simplify Group factors with the same base Laws 5 and 4 Law 3 Laws 1 and 2

10. 10 Rules for Working with Exponents Two additional laws that are useful in simplifying expressions with negative exponents.

11. 11 Example 5 – Simplifying Expressions with Negative Exponents Eliminate negative exponents and simplify each expression. (a) (b)

12. 12 Example 5 – Solution (a) We use Law 7, which allows us to move a number raised to a power from the numerator to the denominator (or vice versa) by changing the sign of the exponent. Law 7 Law 1

13. 13 Example 5 – Solution (b) We use Law 6, which allows us to change the sign of the exponent of a fraction by inverting the fraction. Law 6 Laws 5 and 4 cont’d

14. 14 Scientific Notation For instance, when we state that the distance to the star Proxima Centauri is 4  10 13 km, the positive exponent 13 indicates that the decimal point should be moved 13 places to the right:

15. 15 Scientific Notation When we state that the mass of a hydrogen atom is 1.66  10 –24 g, the exponent –24 indicates that the decimal point should be moved 24 places to the left:

16. 16 Radicals The symbol means “the positive square root of.” Thus = b means b 2 = a ; b  0; a  0

17. 17 Radicals

18. 18 Example 8 – Simplifying Expressions Involving nth Roots (a) (b) Property 4: Property 1: Factor out the largest cube Property 1: Property 5, Property 5:

19. 19 Example 9 – Combining Radicals (a) (b) If b > 0, then Property 1: Factor out the largest squares Property 1: Property 5, b > 0 Distributive property

20. 20 Rational Exponents A rational exponent or a fractional exponent such as a 1/3, is represented by a radical. To give meaning to the symbol a 1/n in a way that is consistent with the Laws of Exponents, we would have to have (a 1/n ) n = a (1/n)n = a 1 = a The nth root a 1/n =

21. 21 Example 11 – Using the Laws of Exponents with Rational Exponents (a) a 1/3 a 7/3 = a 8/3 (b) = a 2/5 + 7/5 – 3/5 = a 6/5 (c) (2a 3 b 4 ) 3/2 = 2 3/2 (a 3 ) 3/2 (b 4 ) 3/2 = ( ) 3 a 3(3/2) b 4(3/2) = 2 a 9/2 b 6 Law 1, Law 2: Law 1: a m b n = a m +n Law 4: (abc) n = a n b n c n Law 3: (a m ) n = a mn

22. 22 Example 11 – Using the Laws of Exponents with Rational Exponents (d) Law 1, Law 2 Law 3 Laws 5, 4, and 7

23. 23 Rationalizing the Denominator It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator. If the denominator is of the form, we multiply numerator and denominator by. In doing this we multiply the given quantity by 1, so we do not change its value. For instance,

24. 24 Example 13 – Rationalizing Denominators (a) (b) (c)