1.2 – Day 1 Exponents and Radicals

2 Objectives ► Integer Exponents ► Rules for Working with Exponents ► Scientific Notation ► Radicals ► Rational Exponents ► Rationalizing the Denominator

3 Integer Exponents

4 A product of identical numbers is usually written in exponential notation. For example, 5  5  5 is written as 5 3. In general, we have the following definition:

5 Example 1 – Exponential Notation Evaluate: (a) (b) (–3) 4 (c) –3 4

6 Integer Exponents

7 Example 2 – Zero and Negative Exponents Evaluate: (a) (b) (c)

8 Rules for Working with Exponents

9 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.

10 Example 4 – Simplifying Expressions with Exponents Simplify: (a) (2a 3 b 2 )(3ab 4 ) 3 (b)

11 Rules for Working with Exponents Here are two additional laws that are useful in simplifying expressions with negative exponents:

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

13 Scientific Notation

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

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 Example 6 – Changing from Decimal to Scientific Notation Write each number in scientific notation. (a) 56,920(b)

17 Example 7 – Calculating with Scientific Notation Evaluate: (a) (b)

18 Practice: p #1, 15-18, 35-51o, 77-81o, 83-87o, 93-97o, 103

– Day 2 Exponents and Radicals

20 Objectives ► Integer Exponents ► Rules for Working with Exponents ► Scientific Notation ► Radicals ► Rational Exponents ► Rationalizing the Denominator

21 Radicals

22 Radicals We know what 2 n means whenever n is an integer. To give meaning to a power, such as 2 4/5, whose exponent is a rational number, we need to discuss radicals. The symbol means “the positive square root of.” Thus, = b means b 2 = a and b  0 Since a = b 2  0, the symbol makes sense only when a  0. For instance, = 3 because 3 2 = 9 and 3  0

23 Radicals Square roots are special cases of nth roots. The nth root of x is the number that, when raised to the nth power, gives x.

24 Radicals

25 Example 8 – Simplifying Expressions Involving nth Roots Simplify: (a)(b)

26 Example 9 – Combining Radicals Simplify: (a)(b) If b > 0, then

27 Rational Exponents

28 Rational Exponents To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a 1/3, we need to use radicals. 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 So by the definition of nth root, a 1/n =

29 Rational Exponents In general, we define rational exponents as follows.

30 Example 11 – Using the Laws of Exponents with Rational Exponents Simplify: (a) (b)

31 Example 11 – Using the Laws of Exponents with Rational Exponents Simplify: (c) (2a 3 b 4 ) 3/2 (d)

32 Rationalizing the Denominator

33 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,

34 Example 13 – Rationalizing Denominators Simplify: (a) (b) (c)

35 Practice: p #7-13o, 19-33o, 53, 55, 61-75o, 89, 91, 101, 102