1.2 Objectives
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1.2 Objectives • Integer Exponents • Rules for Working with Exponents • Scientific Notation • Radicals • Rational Exponents • Rationalizing the Denominator
Example 1 – Exponential Notation • (a) • (b) (–3)4 = (–3) (–3) (–3) (–3) • = 81 • (c) –34 = –(3 3 3 3) • = –81
Integer Exponents • A product of identical numbers is usually written in exponential notation. For example, 5 5 5 is written as 53.
Example 2 – Zero and Negative Exponents • (a) • (b) • (c) • = 1
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.
Example 4 – Simplifying Expressions with Exponents • Simplify: • (2a3b2)(3ab4)3 • Solution: • (2a3b2)(3ab4)3 = (2a3b2)[33a3(b4)3] • = (2a3b2)(27a3b12) • = (2)(27)a3a3b2b12 • = 54a6b14 • Law 4: (ab)n = anbn • Law 3: (am)n = amn • Group factors with the same base • Law 1: ambn = am + n
Example 4 • Simplify • Laws 5 and 4 • Law 3 • Group factors with the same base • Laws 1 and 2
Rules for Working with Exponents • Two additional laws that are useful in simplifying expressions with negative exponents.
Example 5 – Simplifying Expressions with Negative Exponents • Eliminate negative exponents and simplify each expression. • (a) • (b)
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
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
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:
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:
Radicals • The symbol means “the positive square root of.” Thus • = b means b2 = a ; b 0; a 0
Example 8 – Simplifying Expressions Involving nth Roots • (a) • (b) • Factor out the largest cube • Property 1: • Property 4: • Property 1: • Property 5, • Property 5:
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
Rational Exponents • A rational exponent or a fractional exponent such as a1/3, is represented by a radical. 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 • The nth root a1/n=
Example 11 – Using the Laws of Exponents with Rational Exponents • (a) a1/3a7/3 = a8/3 • (b) = a2/5 + 7/5 – 3/5 • = a6/5 • (c) (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
Example 11 – Using the Laws of Exponents with Rational Exponents • (d) • Laws 5, 4, and 7 • Law 3 • Law 1, Law 2
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,
Example 13 – Rationalizing Denominators • (a) • (b) • (c)
