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

2. Integer Exponents
A product of identical numbers is usually written in exponential notation. For example, 5  5  5 is written as 53. In general, we have the following definition.

3. Example 1 – Exponential Notation
(b) (–3)4 = (–3)  (–3)  (–3)  (–3)
= 81
(c) –34 = –(3  3  3  3)
= –81

4. Integer Exponents

5. Example 2 – Zero and Negative Exponents
(b)

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. Example 4 – Simplifying Expressions with Exponents
(a) (2a3b2)(3ab4)3
(b)
Solution:
(a) (2a3b2)(3ab4)3 = (2a3b2)[33a3(b4)3]
= (2a3b2)(27a3b12)
= (2)(27)a3a3b2b12
Law 4: (ab)n = anbn
Law 3: (am)n = amn
Group factors with the same base

8. Example 4 – Solution = 54a6b14 (b) cont’d Law 1: ambn = am + n
Laws 5 and 4
Law 3
Group factors with the same base
Laws 1 and 2

9. Rules for Working with Exponents
We now give two additional laws that are useful in simplifying expressions with negative exponents.

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

11. 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

12. Example 5 – Solution
cont’d
(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

13. Scientific Notation
For instance, when we state that the distance to the star Proxima Centauri is 4  1013 km, the positive exponent 13 indicates that the decimal point should be moved 13 places to the right:

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

15. Example 6 – Changing from Decimal to Scientific Notation
Write each number in scientific notation.
(a) 56,920
(b)
Solution:
(a) 56,920 =  104
(b) = 9.3  10–5

16. Radicals
We know what 2n means whenever n is an integer. To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals.
The symbol means “the positive square root of.” Thus
= b means b2 = a and b  0
Since a = b2  0, the symbol makes sense only when a  0. For instance,
= because = 9 and  0

17. 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.

18. Radicals

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

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

21. Rational Exponents
To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals. To give meaning to the symbol a1/n in a way that is consistent with the Laws of Exponents, we would have to have
(a1/n)n = a(1/n)n = a1 = a
So by the definition of nth root,
a1/n =

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

23. Example 11 – Using the Laws of Exponents with Rational Exponents
(a) a1/3a7/3 = a8/3
(b) (2a3b4)3/2 = 23/2(a3)3/2(b4)3/2
= ( )3a3(3/2)b4(3/2)
= a9/2b6
Law 1: ambn = am +n
Law 1, Law 2:
Law 4: (abc)n = anbncn
Law 3: (am)n = amn

24. 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,

25. Example 13 – Rationalizing Denominators
(b)