Section A.7 and A.10
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1 Section A.7 and A.10 nth Roots,,, & Math Precalculus I Roots, Exponents, Section A.7 and A.10
2 A.10 nth Roots & A.7 Solve: 3 5 2x 4 < 7 Roots, Exponents, Section A.7 and A.10
3 A.10 nth Roots & A.7 Solve: 3 5 2x 4 < 7 Ans: (, 1) (3, ) 3 5 2x 4 < 7 5 2x 4 < 10 2x 4 > 2 So 2x 4 > 2 or 2x 4 < 2. Roots, Exponents, Section A.7 and A.10
4 A.10 nth Roots & A.7 Our first exam will be Friday during class time in this room. It will cover the Appendix. I will do a review on Wednesday. Be sure to check out the review sheet and exam cover sheet I will post on my web site later today. Roots, Exponents, Section A.7 and A.10
5 Definitions Roots and Radicals Square of a number: Result of multiplying a number by itself 3 2 = (3)(3) = 9 Roots, Exponents, Section A.7 and A.10
6 Definitions Roots and Radicals Square of a number: Result of multiplying a number by itself 3 2 = (3)(3) = 9 Square Root of a number: Number you square to get the given number 9 = 3 since 3 2 = 9 Roots, Exponents, Section A.7 and A.10
7 Definitions Roots and Radicals Square of a number: Result of multiplying a number by itself 3 2 = (3)(3) = 9 Square Root of a number: Number you square to get the given number 9 = 3 since 3 2 = 9 n th Root of a number: Number raised to the n th power to get the given number 5 32 = 2 since ( 2) 5 = 32 Roots, Exponents, Section A.7 and A.10
8 Definitions Roots and Radicals Square of a number: Result of multiplying a number by itself 3 2 = (3)(3) = 9 Square Root of a number: Number you square to get the given number 9 = 3 since 3 2 = 9 n th Root of a number: Number raised to the n th power to get the given number 5 32 = 2 since ( 2) 5 = 32 In general n a = b means a = b n Roots, Exponents, Section A.7 and A.10
9 Definitions Roots and Radicals Square of a number: Result of multiplying a number by itself 3 2 = (3)(3) = 9 Square Root of a number: Number you square to get the given number 9 = 3 since 3 2 = 9 n th Root of a number: Number raised to the n th power to get the given number 5 32 = 2 since ( 2) 5 = 32 In general n a = b means a = b n n is index, a is radicand, is radical symbol Roots, Exponents, Section A.7 and A.10
10 Properties Roots and Radicals Roots, Exponents, Section A.7 and A.10
11 Properties Roots and Radicals If n 2, m 2 and the radicals are defined, then Roots, Exponents, Section A.7 and A.10
12 Properties Roots and Radicals If n 2, m 2 and the radicals are defined, then n ab = n a n b Roots, Exponents, Section A.7 and A.10
13 Properties Roots and Radicals If n 2, m 2 and the radicals are defined, then n ab = n a n b n a n a b = n b Roots, Exponents, Section A.7 and A.10
14 Properties Roots and Radicals If n 2, m 2 and the radicals are defined, then n ab = n a n b n a n a b = n b n a m = ( n a ) m Roots, Exponents, Section A.7 and A.10
15 Properties Roots and Radicals If n 2, m 2 and the radicals are defined, then n ab = n a n b n a n a b = n b n a m = ( n a ) m n a n = a if n 3 and n is odd = a if n 2 and n is even Roots, Exponents, Section A.7 and A.10
16 How to simplify radicals: Roots, Exponents, Section A.7 and A.10
17 How to simplify radicals: Remove any perfect roots from the radicand Roots, Exponents, Section A.7 and A.10
18 How to simplify radicals: Remove any perfect roots from the radicand No fractions under the radicand Roots, Exponents, Section A.7 and A.10
19 How to simplify radicals: Remove any perfect roots from the radicand No fractions under the radicand No radicals in the denominator Roots, Exponents, Section A.7 and A.10
20 How to simplify radicals: Remove any perfect roots from the radicand No fractions under the radicand No radicals in the denominator Examples Roots, Exponents, Section A.7 and A.10
21 Roots, Exponents, Section A.7 and A.10
22 Put the radical on one side and then undo by exponentiating. Roots, Exponents, Section A.7 and A.10
23 Put the radical on one side and then undo by exponentiating. Solve x + 4 = 5 Roots, Exponents, Section A.7 and A.10
24 Put the radical on one side and then undo by exponentiating. Solve x + 4 = 5 Watch out! Incorrect thinking sometimes can lead to a correct solution. Roots, Exponents, Section A.7 and A.10
25 Solve 12 x = x Roots, Exponents, Section A.7 and A.10
26 Solve 12 x = x Be sure the solution is in the domain of the original equation. Roots, Exponents, Section A.7 and A.10
27 Use the rules of exponents as well as Roots, Exponents, Section A.7 and A.10
28 Use the rules of exponents as well as a m/n = n a m = ( n a ) m Roots, Exponents, Section A.7 and A.10
29 Use the rules of exponents as well as a m/n = n a m = ( n a ) m Examples: Simplify 3 x 2 x 4 x 3 Roots, Exponents, Section A.7 and A.10
30 Use the rules of exponents as well as a m/n = n a m = ( n a ) m Examples: Simplify 3 x 2 x 4 x 3 ( 4x 1 y 1/3) 2/3 ( x 1 y ) 3/2 Roots, Exponents, Section A.7 and A.10
31 Imagine the solution to the equation x = 0. Roots, Exponents, Section A.7 and A.10
32 Imagine the solution to the equation x = 0. In 1572 Italian Rafael Bombelli defined a new number: i = 1 Roots, Exponents, Section A.7 and A.10
33 Imagine the solution to the equation x = 0. In 1572 Italian Rafael Bombelli defined a new number: i = 1 At the time, such numbers were regarded by some as useless. For example, René Descartes called them imaginary in 1637 because they appeared to be fictitious. Roots, Exponents, Section A.7 and A.10
34 Uses of Imaginary numbers are used in signal processing Roots, Exponents, Section A.7 and A.10
35 Uses of Imaginary numbers are used in signal processing control theory Roots, Exponents, Section A.7 and A.10
36 Uses of Imaginary numbers are used in signal processing control theory electromagnetism Roots, Exponents, Section A.7 and A.10
37 Uses of Imaginary numbers are used in signal processing control theory electromagnetism fluid dynamics Roots, Exponents, Section A.7 and A.10
38 Uses of Imaginary numbers are used in signal processing control theory electromagnetism fluid dynamics quantum mechanics Roots, Exponents, Section A.7 and A.10
39 Uses of Imaginary numbers are used in signal processing control theory electromagnetism fluid dynamics quantum mechanics cartography Roots, Exponents, Section A.7 and A.10
40 For example the models that describe how AC current flows through wires use imaginary numbers. Roots, Exponents, Section A.7 and A.10
41 For example the models that describe how AC current flows through wires use imaginary numbers. Here is a model for the propagation of a plane wave along the x-axis as a function of time: Roots, Exponents, Section A.7 and A.10
42 For example the models that describe how AC current flows through wires use imaginary numbers. Here is a model for the propagation of a plane wave along the x-axis as a function of time: Ψ(x, t) = 1 ( ) 2π θ e i( 2π x ωt) λ d 2π 2π λ λ where Ψ is the wave function λ is the wavelength θ is a characteristic of the particular wave ω is frequency t is time Roots, Exponents, Section A.7 and A.10
43 Powers of i Roots and Radicals i = 1 Roots, Exponents, Section A.7 and A.10
44 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 Roots, Exponents, Section A.7 and A.10
45 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 i 3 = i 2 i = 1 i = i Roots, Exponents, Section A.7 and A.10
46 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 i 3 = i 2 i = 1 i = i i 4 = i 2 i 2 = 1 1 = 1 Roots, Exponents, Section A.7 and A.10
47 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 i 3 = i 2 i = 1 i = i i 4 = i 2 i 2 = 1 1 = 1 i 5 = i 4 i = 1 i = i Roots, Exponents, Section A.7 and A.10
48 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 i 3 = i 2 i = 1 i = i i 4 = i 2 i 2 = 1 1 = 1 i 5 = i 4 i = 1 i = i i 6 = i 4 i 2 = 1 ( 1) = 1 Roots, Exponents, Section A.7 and A.10
49 Powers of i Roots and Radicals i = 1 i 2 = ( 1 ) 2 = 1 i 3 = i 2 i = 1 i = i i 4 = i 2 i 2 = 1 1 = 1 i 5 = i 4 i = 1 i = i i 6 = i 4 i 2 = 1 ( 1) = 1 and so on... Roots, Exponents, Section A.7 and A.10
50 Complex numbers have the form a + bi where: a and b are real numbers Roots, Exponents, Section A.7 and A.10
51 Complex numbers have the form a + bi where: a and b are real numbers i is the imaginary unit, 1 Roots, Exponents, Section A.7 and A.10
52 Complex numbers have the form a + bi where: a and b are real numbers i is the imaginary unit, 1 a is the real part of the complex number Roots, Exponents, Section A.7 and A.10
53 Complex numbers have the form a + bi where: a and b are real numbers i is the imaginary unit, 1 a is the real part of the complex number b is the imaginary part of the complex number Roots, Exponents, Section A.7 and A.10
54 Complex numbers have the form a + bi where: a and b are real numbers i is the imaginary unit, 1 a is the real part of the complex number b is the imaginary part of the complex number Roots, Exponents, Section A.7 and A.10
55 Complex numbers have the form a + bi where: a and b are real numbers i is the imaginary unit, 1 a is the real part of the complex number b is the imaginary part of the complex number z = a + bi z = a bi is the complex conjugate Roots, Exponents, Section A.7 and A.10
56 Operations with Tip: Add like normal and think of the i as a variable like x. Examples: Roots, Exponents, Section A.7 and A.10
57 Operations with Tip: Add like normal and think of the i as a variable like x. Examples: (2 3i)(4 + i) Roots, Exponents, Section A.7 and A.10
58 Operations with Tip: Add like normal and think of the i as a variable like x. Examples: (2 3i)(4 + i) 4 3i Roots, Exponents, Section A.7 and A.10
59 Operations with Tip: Add like normal and think of the i as a variable like x. Examples: (2 3i)(4 + i) 4 3i i Roots, Exponents, Section A.7 and A.10
60 Square roots of negative numbers Define the principal square root of a negative number as follows: Roots, Exponents, Section A.7 and A.10
61 Square roots of negative numbers Define the principal square root of a negative number as follows: N = Ni Roots, Exponents, Section A.7 and A.10
62 Square roots of negative numbers Define the principal square root of a negative number as follows: N = Ni Examples: Roots, Exponents, Section A.7 and A.10
63 Remember, at the first lecture, I proved that 1 = 2? Let s prove that false by proving that 1 = 1 instead... Roots, Exponents, Section A.7 and A.10
64 Start working on the review problems Roots, Exponents, Section A.7 and A.10
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