Algebraic Expressions with a Technical Example 1 Evaluating Algebraic Expressions Evaluating Expressions Using Technical Applications

Algebraic Expressions with a Technical Example 2 Scientists, engineers and technicians need, develop, and use mathematics to explain, describe, and predict what nature, processes, and equipment do. Many times the math they use is the math that is taught in ALGEBRA 1! Technical Application

Algebraic Expressions with a Technical Example 3 1) To simplify algebraic expressions by using the rules for order operations to evaluate algebraic expressions. There are 3 Objectives of this presentation: To evaluate algebraic expressions involving multiplication and division of real numbers. 2) To use algebra 1 to help understand a technical application. 3)

Algebraic Expressions with a Technical Example 4 The rules for dividing real numbers involves the mathematical concept of RECIPROCALS. TWO EXAMPLES (-27)= 1 3 The fraction one third is the reciprocal of 3 Divide –27 by 3 is the same as multiply –27 by the reciprocal of 3. d= 1 2 )(b == d 1 2 )()( b 1 ( 2b1 ))()( 11dd 2 )( b (a) Specific Situation = = 3 The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. () (b) General Situation the fraction 1 )( b is the reciprocal of b A common symbol technicians, scientists and engineers use for multiplication. Evaluating Expressions d 1 2 )()( b 1 ( 2b1 ))()( 11d

Algebraic Expressions with a Technical Example 5 Evaluating Expressions 4 EASY PRACTICE PROBLEMS The fraction one fourth Dividing 1 by 4 is the same as multiplying 1 by the fraction 4 = 1 ? = is the reciprocal of the number Dividing 1 by 4 is the same as multiplying 1 by the reciprocal of the number is the reciprocal of the number 4 (a) Divide the number 1 by the number 4 The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number.

Algebraic Expressions with a Technical Example 6 9 = 1 ? 0.111= 1 9 (b) The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. Evaluating Expressions 4 EASY PRACTICE PROBLEMS (continued)

Algebraic Expressions with a Technical Example 7 (c) If the values of d and b are 5 and 36 respectively, what is the value of the following algebraic expression? = (0.10) (0.14) 1 10 )( 5 36 )( ? = )()( 36 1 (d 1 10 ))( b 1 = = = d= 1 10 )(b )( 1 b are reciprocals of each other. and the variable The variable b The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. Evaluating Expressions 4 EASY PRACTICE PROBLEMS (continued)

Algebraic Expressions with a Technical Example 8 Sometimes it is fun in Algebra to use a letter from the Greek alphabet as well as letters like “d” and “b”. Try the following problem using the Greek letter LAMBDA (d) the Greek letter Lambda = ? = when b equals 36 and d equals 5. 10b )( d 1 The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. Evaluating Expressions 4 EASY PRACTICE PROBLEMS (continued)

Algebraic Expressions with a Technical Example 9 The rules for dividing real numbers involve the mathematical concept of reciprocals. The division by a number is defined as multiplying by the reciprocal or multiplicative inverse of the number. 4 Easy Practice Problems (continued) (d) when b equals 36 and d equals 5. = = 10b )( d 1 5 ()36 =5360() 1800 = ? (d) 10 )( b 1 the Greek letter Lambda = ? = 10b )( d 1 What is the reciprocal of ? 10b )( 1 Evaluating Expressions

Algebraic Expressions with a Technical Example 10 Technicians, scientists and engineers use Algebra all the time. When using algebra, they also like to use combinations of letters and numbers as algebra symbols. = 10b )( d 1 The previous example problem used the Greek letter lambda as well as the letters “d” and “b”. A technician might use this algebraic expression with lambda and the letters “d” and “b” replaced by symbols that are combinations of letters and numbers. =d 1 10b )( 1 1  Evaluating Expressions = 10b )( d 1

Algebraic Expressions with a Technical Example 11 Technicians, scientists and engineers use Algebra all the time. However, they also like to use combinations of letters and numbers as algebra symbols. = 10 )( 1 d 1 b 1 when b 1 equals 36 and d 1 equals 5. = ? = ( 10 ) )( 1 b 1 d 1 5 ()36   =1800 PRACTICE PROBLEM  == = d1d1 10)b1b1 ( = ?  when b 1 equals 36 and d 1 equals 5. d1d () b1b () =0.72 What is the reciprocal of ? 10d )( 1 Evaluating Expressions

Algebraic Expressions with a Technical Example 12 Scientists, technicians and engineers also use algebraic symbols that are combinations of letters and numbers because they often work with the same algebraic expression but substitute different numbers. b 1 = 10 )( 1 d 1 = ? = ( 10 ) )( 1 b 1 d 1 5 ()36  =1800  = Let b 1 equal 36 and d 1 equal 5. TWO QUICK EXAMPLE PROBLEMS (1) (2) These examples use the following algebraic expression; This time, let b 1 equal 35 and d 1 equal 5. ? = ( 10 ) )( 1 b 1 d 1 5 ()35=1750 substitute different numbers.

Algebraic Expressions with a Technical Example 13 = ? (1) (2) 0.7 Sometimes an engineer, scientist or technician may select symbols that are similar when the algebraic expressions are different. 2 EASY EXAMPLE PROBLEMS In both problems b 1 equals 35 and d 1 equals 5. This often happens when there is a connection between the answers after the expressions have been evaluated. =  10 ) ( d 1 b = ( 10 ) 7 1 = ( 50( ) 35 ) =  10 ) ( d 1 b 1 = ? 1 = ( 35( ) 50 )= ( 7 ) 10 =  10 ) ( d 1 b 1 =  10 ) ( d 1 b 1 =  10 ) ( d 1 b 1

Algebraic Expressions with a Technical Example 14 = ? (1) (2) 0.7 =  10 ) ( d 1 b = ( 10 ) 7 1 = ( 50( ) 35 ) =  10 ) ( d 1 b 1 = ? 1 = ( 35( ) 50 )= ( 7 ) 10   ()() = (1.43) (0.7) = 1.00 What is the connection between and ?     is the reciprocal of   is the reciprocal of? Or, if you wish 1 Evaluating Expressions

Algebraic Expressions with a Technical Example 15 THREE QUICK REVIEW QUESTIONS to see what we remember 1)   If = 1.43 and = 0.7   What is the connection betweenand? One is the reciprocal of the other. 2)What is the answer if you multiply reciprocals together? You always get the number 1 as the answer. 3) Try this with a calculator. Is there a problem? What is the reciprocal of ? 10d )( 1 d 1 )( 1

Algebraic Expressions with a Technical Example 16 What do you think? 1) 2)Do all fractions have reciprocals? Why/Why not? Is one half the reciprocal of 2? Why/why not? 3)Two of the most popular manufacturers of calculators (TI and HP) have a different style (way to do calculation) for getting answers to multiplication and division problems. One of them was developed with a knowledge (use) of reciprocals in mind. Which one is it? Why? 4)Use the Web (if you have to) and examine the arrangement of the scales on a slide rule. One of those scales is know as the reciprocal scale. Which one is it and why is it named so?

Algebraic Expressions with a Technical Example 17 Objective: To simplify algebraic expressions by using the rules for order operations to evaluate algebraic expressions Evaluating Expressions Simplifying Expressions2)

Algebraic Expressions with a Technical Example 18 Perform operations within parentheses first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. TWO EXAMPLES = = )( = )( )( = )( = = (a) (b) Rules = Simplifying Expressions

Algebraic Expressions with a Technical Example 19 Perform operations within parenthesis first. Add (subtract) in order from left to right. TWO SIMPLE EXAMPLES (103)14-+10= = = =(103) (a) ? = = = Multiply (divide) in order from left to right. Perform operations within parentheses first. Add (subtract) in order from left to right. Rules that help the Simplification of Expressions Simplifying Expressions

Algebraic Expressions with a Technical Example 20 = )( = )( )( = )( = Another way that technicians, scientists and engineers often simplify this type of algebraic expression. = )( )( 1 (9-4) (4 9) = )( = )( = Rules used? Perform operations within parentheses first. Multiply (divide) in order from left to right. (b) = Rule to use first? Perform operations within parenthesis Multiply (divide) in order from left to right. Simplifying Expressions

Algebraic Expressions with a Technical Example 21 TWO GENERALIZED EXAMPLES = )( - d 1 b )( 1 (d – b) (b d) Perform operations within parentheses first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. (a) )( For the previous problem, b was equal to 4 and d was equal to 9 Rules that help the Simplification of Expressions Simplifying Expressions

Algebraic Expressions with a Technical Example 22 = )( - n2n2 1 n1n )( (n 2 – n 1 ) 10 1 (n 1 n 2 ) This time, the symbol n 1 replaces the letter b, and the symbol n 2 replaces the letter d. (b) = )( - d 1 b )( (d – b) 10 1 (b d) Technical workers often use different symbol combinations for the letters b and d. Simplifying Expressions

Algebraic Expressions with a Technical Example 23 Simplification of a new expression: )( - n2n2 1 n1n This time, let n 1 equal 2 and n 2 equal 3 = ?0.014 )( = = )( )( (9 -4) 10 1 (4 9) 22 = ? Note: The calculations inside the parentheses were completed before multiplying by one tenth. )( = Simplifying Expressions

Algebraic Expressions with a Technical Example 24 Perform operations within parenthesis first. Reciprocal Expressions 10 [ ] 1 = 10 [] = [] =0.10 THREE EASY EXAMPLE PROBLEMS a) b) [ ] 1 = 10 [ ] 1 = There is nothing to do inside this parenthesis. There is something to do inside this parenthesis. Multiply (divide) in order from left to right (2) +2 [ ] 1 = Rules Perform operations within parentheses first. Add (subtract) in order from left to right. Simplifying Expressions

Algebraic Expressions with a Technical Example 25 These two expressions are same. )( - n2n2 1 n1n [] 1 = )( - n2n2 1 n1n [] c) )( [ ] 1 = = [ ] 1 )( [] A typical reciprocal (inverse) expression used in technology. 10 [ ] 1 = This version is popular in technical applications because it takes up less space on a piece of paper and is easier to type on a computer. Rules Perform operations within parentheses first. Reciprocal Expressions Simplifying Expressions

Algebraic Expressions with a Technical Example 26 What is the value of this expression when n 1 equals 2 and n 2 equals 3? )( - n2n2 1 n1n [] 1 = )( - n2n2 1 n1n [] PRACTICE PROBLEM Rules Perform operations within parentheses first. Reciprocal Expressions Simplifying Expressions

Algebraic Expressions with a Technical Example 27 Note: 2 2 = 2 times 2 = = 3 times 3 = 9 = = Perform operations within parenthesis first = )( [ ] )( 10 1 (9 -4) (4 9) [] = = )( [ ] )( [] [] 71.4 n 1 equals 2 and n 2 equals 3 The calculation of the inverse is the last thing done )( - n2n2 1 n1n [] () [] Note: = () is the number ) 2) Simplifying Expressions

Algebraic Expressions with a Technical Example 28 THREE QUICK REVIEW QUESTIONS. 1)What are, in the correct order of use, the rules for simplifying algebraic expressions? 2)What is another way to write the following algebraic expression? ) Perform operations within parentheses first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. )( - n2n2 1 n1n1 1 )( (n 2 – n 1 ) (n 1 n 2 ) = What is (b) (a) the inverse of ? () 1 () the reciprocal of the the number 71.4? Simplifying Expressions

Algebraic Expressions with a Technical Example 29 What do you think? 1) (a) Is the inverse of a number always the same as the reciprocal of that number? Why/Why not? Are the two algebraic expressions show below equal? Why/why not? -2-2 )( - n2n2 1 n1n [] 2) (b) )( - n2n2 1 n1n [] 1 2 Simplifying Expressions

Algebraic Expressions with a Technical Example 30 Objective: To use algebraic expressions to describe and understand a technical application. Simplifying Expressions 3)Technology Application

Algebraic Expressions with a Technical Example 31 Using Algebra in a Technical Application Consider the feasibility of using a hydrogen laser beam for application as a welding tool outside the space station. A laser beam is made when the an electron from many of the same type of atoms moves back from the same outer orbit to the same orbit closer to the atom’s nucleus. Laser welding tool EXCITATION SOURCE Atoms emit light as photons after excitation Some travel parallel to the laser tube and bounce off the mirrored ends Monochromatic light (LASER LIGHT) leaves the partially mirrored end SIMPLIFIED LASER LASER TUBE

Algebraic Expressions with a Technical Example 32 Using Algebra in a Technical Application Consider the feasibility of using a hydrogen laser beam for application as a welding tool outside the space station. A laser beam is made when the an electron from many of the same types of atoms moves back from the same outer orbit to the same orbit closer to the atom’s nucleus. Atom’s nucleus 2 st orbit out from nucleus (n 2 ) 3 nd orbit out from nucleus (n 3 ) If, for example, electrons move from 3 rd orbit to second orbit, a single color light will be emitted. A photon of light is emitted If the same electrons from enough of the same type of atoms go through this process, we may see a collection of light waves as a beam of a specific colored light.

Algebraic Expressions with a Technical Example 33 Technology Example: In 1875 Balmer determined a mathematical relationship that is used today to predict the wavelength of laser light generated by atomic gases. )( - n3n3 1 n2n [] = Balmer’s wavelength value in nanometers A technician, engineer, or scientist will know what color the light is if the length of the wave is known. The length of one cycle of a wave is its wavelength.  This wave is 4½ wavelengths long. Shortest distance to same point on the wave is one wavelength distance Courtesy of NASA

Algebraic Expressions with a Technical Example 34 = = = )( [ ] = = )( [ ] )( [] [] 670 nanometers )( - n3n3 1 n2n [] = Symbol people use to represent the wavelength value in nanometers = ? If the light beam for a hydrogen gas laser is to be generated when one electron from many individual hydrogen atoms in the gas move from the 3 rd orbit to the 2 nd orbit, what color is the light that is observed. Technology Application Problem Statement )( 91 1 (9 -4) (4 9) []

Algebraic Expressions with a Technical Example 35 What color do you thing the light wave is? Technology Application Problem Statement = )( [] =)( [] )( - n2n2 1 n1n [] = wavelength value = = [] 670 nanometers If the light beam for a hydrogen gas laser is to be generated when an electron moves from the 3 rd orbit to the 2 nd orbit of many individual hydrogen atoms in the gas, what color is the light that is observed? The hydrogen laser will emit light of 670 nanometers

Algebraic Expressions with a Technical Example 36 What color do you thing the light wave is? Technology Application Problem Statement If the light beam for a hydrogen gas laser is to be generated when an electron moves from the 3 rd orbit to the 2 nd orbit of many individual hydrogen atoms in the gas, what color is the light that is observed? The hydrogen laser will emit light of 670 nanometers The 670 nm light emitted from the hydrogen laser will be slightly to the right of the color at 657 nm 657 nm 486 nm 434 nm 410 nm

Algebraic Expressions with a Technical Example 37 THREE QUICK REVIEW QUESTIONS 1)   If =.0015 and = 670   What is the relationship betweenand? They are complements. 2)What is the answer if you multiply complements together? You always get the number 1 as the answer. 3) His last name was Balmer and he worked out his formula in What was the last name of the person who determined the mathematical model that is used to predict the color of light caused by electrons moving from one orbit to another? Try this with a calculator. What’s the problem? Technology Example:

Algebraic Expressions with a Technical Example 38 What do you think? 1) 2)Can you make a laser beam by focusing sun light through a prism? Why/ Why not? Do you think Balmer figured out his mathematical relationship so that he could understand the Bohr atomic model? Why/Why not? Technology Example: 3)Does the Balmer model predict that a laser beam will occur if electrons go from the 2nd orbital to the first orbital of an atom? Why/why not.

Algebraic Expressions with a Technical Example 39