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1.2 Objectives

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

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1.2 Objectives

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  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 • (a) • (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 • (a) • (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 • Simplify: • (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 • cont’d • = 54a6b14 • (b) • 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 starProxima 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 1.66  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) 0.000093 • Solution: • (a) 56,920 = 5.692  104 • (b) 0.000093 = 9.3  10–5

  16. Radicals • We know what 2nmeans 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, • = 3 because 32 = 9 and 3  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 • (a) • (b) • Factor out the largest cube • Property 1: • Property 4: • Property 1: • Property 5, • Property 5:

  20. Example 9 – Combining Radicals • (a) • (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/nin 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) • = 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 • (a) • (b)

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