3 Simplifying Radicals Review and Radicals as Exponents
4 A radical expression contains a root, which can be shown using the radical symbol, . The root of a number x is a number that, when multiplied by itself a given number of times, equals x.For Example: 2 4 , 3 8 , 𝑛 𝑥Simplifying RadicalsBasic Review
5 Simplifying Radicals Steps Use a factor tree to put the number in terms of its prime factors.Group the same factor in groups of the number on the outside.Merge those numbers into 1 and place on the outside.Multiply the numbers outside together and the ones left on the inside together.∗2∗2∗3∗3∗3∗5 2∗
6 To add and/or subtract radicals you must first Simplify them, then combine like radicals. Ex: − 2 502 2∗3∗ ∗2∗3 − 2 5∗5∗2−5 2 22 2 3 −2 2 2Simplifying RadicalsAdding and Subtracting
7 Square Roots as Exponents ∗3∗3∗3 3*3 9Please put this in your calculator. What did you get? = 9
8 Bellringer 9/24/14Please get the calculator that has your seat number on it, if there isn’t one please see me!Simplify: 4 32Rewrite as an exponent and solve on your calculator:=2 4 2=4
9 Exponent Rules and Imaginary Numbers - with multiplying and dividing square roots if we have time
10 Imaginary Numbers Can you take the square root of a negative number? Ex: 2 −4 → what number times itself ( 𝑥 2 ) gives you a negative 4?Can u take the cubed root of a negative number?Ex: 3 −8 → what number times itself, and times ( 𝑥 3 ) itself again gives you a negative 8?The imaginary unit i is used to represent the non-real value, 2 −1 .An imaginary number is any number of the form bi, where b is a real number, i = 2 −1 , and b ≠ 0.
11 a0 = 1 𝑎 ( −𝑚 𝑛 ) 1 = 1 𝑎 ( 𝑚 𝑛 ) Exponent Rules Zero Exponent PropertyNegative Exponent PropertyA base raised to the power of 0 is equal to 1.a0 = 1A negative exponent of a number is equal to the reciprocal of the positive exponent of the number.𝑎 ( −𝑚 𝑛 ) 1 = 1 𝑎 ( 𝑚 𝑛 )Examples:
12 𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛 𝑎 𝑚 ∗ 𝑎 𝑛 = 𝑎 𝑚+𝑛 Exponent Rules Quotient of Powers PropertyProduct of Powers PropertyTo multiply powers with the same base, add the exponents.𝑎 𝑚 ∗ 𝑎 𝑛 = 𝑎 𝑚+𝑛To divide powers with the same base, subtract the exponents.𝑎 𝑚 𝑎 𝑛 = 𝑎 𝑚−𝑛Examples:
13 ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚∗𝑛 (𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚 Exponent Rules Power of a Power PropertyPower of a Product PropertyTo raise one power to another power, multiply the exponents.( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚∗𝑛To find the power of a product, distribute the exponent.(𝑎𝑏) 𝑚 = 𝑎 𝑚 ∗ 𝑏 𝑚Examples:
14 ( 𝑎 𝑏 ) 𝑚 = 𝑎 𝑚 𝑏 𝑚 Exponent Rules Power of a Quotient PropertyTo find the power of a quotient, distribute the exponent.( 𝑎 𝑏 ) 𝑚 = 𝑎 𝑚 𝑏 𝑚Examples:
17 Roots and Radicals Review The Rules (Properties)MultiplicationDivisionb may not be equal to 0.
18 Roots and Radicals b may not be equal to 0. The Rules (Properties) MultiplicationDivisionb may not be equal to 0.
19 Roots and Radicals Review Examples:MultiplicationDivision
20 Roots and Radicals Review Examples: MultiplicationDivision
21 Intermediate Algebra MTH04 Roots and RadicalsTo add or subtract square roots or cube roots...simplify each radicaladd or subtract LIKE radicals byadding their coefficients.Two radicals are LIKE if they have the same expression under the radical symbol.
23 Complex NumbersAll complex numbers are of the form a + bi, where a and b are real numbers and i is the imaginary unit. The number a is the real part and bi is the imaginary part.Expressions containing imaginary numbers can also be simplified.It is customary to put I in front of a radical if it is part of the solution.
24 Simplifying with Complex Numbers Practice Problem 1𝑖+ 𝑖 3𝑖+ 𝑖∗ 𝑖 2𝑖+ 𝑖∗ −1𝑖−𝑖=0Problem 23 −8 + 2 −83 (−2)(−2)(−2) + 2 (2)(2)(2)(−1)− (−1)− ∗ 2 −1=−2+2𝑖 2 2
26 PracticeWith Sub – simplify, i, complex, exponent rules
27 Bellringer 9/29/14 Is this your classroom? Write all of these questions and your responseIs this your classroom?Should you respect other people’s property and work space?Should you alter Mrs. Richardson’s Calendar?How should you treat the class set of calculators?
28 Review Practice Answers Discuss what to do when there is a substitute
29 Bellringer 9/30/14. EQ- What are complex numbers Bellringer 9/30/14 *EQ- What are complex numbers? How can I distinguish between the real and imaginary parts?1. How often should we staple our papers together?When should we turn in homework and where?When and where should we turn in late work?4. What are real numbers?
30 Let’s Review the real number system! Rational numbersIntegersWhole NumbersNatural NumbersIrrational Numbers
32 Now we have a new number! Complex Numbers Defined. Complex numbers are usually written in the form a+bi, where a and b are real numbers and i is defined as Because does not exist in the set of real numbers I is referred to as the imaginary unit.If the real part, a, is zero, then the complex number a +bi is just bi, so it is imaginary.0 + bi = bi , so it is imaginaryIf the real part, b, is zero then the complex number a+bi is just a, so it is real.a+ 0i =a , so it is real
33 Examples Name the real part of the complex number 9 + 16i? What is the imaginary part of the complex numbers i?
34 Check for understanding Name the real part of the complex number i?What is the imaginary part of the complex numbers i?Name the real part of the complex number 16i?What is the imaginary part of the complex numbers 23?
35 Name the real part and the imaginary part of each. 22.214.171.124.5.
36 Bellringer 10/1/14 *EQ- How can I simplify the square root of a negative number? For Questions 1 & 2, Name the real part and the imaginary part of each.For Questions 3 & 4, Simplify each of the following square roots.
37 Simply the following Square Roots.. How would you take the square root of a negative number??
38 Simplifying the square roots with negative numbers The square root of a negative number is an imaginary number.You know that i =When n is some natural number (1,2,3,…), then
45 A negative number raised to an even power will always be positive Note:A negative number raised to an even power will always be positiveA negative number raised to an odd power will always be negative.
46 How could we make a list of i values? 1−1 =𝑖𝑖∗𝑖= −1 ∗ −1 =−1𝑖 2 ∗𝑖= −1 ∗ −1 =−𝑖(𝑖 2 ) 2 = −1 2 =−1∗−1=1(𝑖 2 ) 2 ∗𝑖= −1 2 ∗ −1 =1∗ −1 = −1 =𝑖(𝑖 2 ) 3 = −1 3 =−1∗−1∗−1=−1