Simplifying Expressions in Algebraic Expressions Applications in Atomic Sciences

Many times the math they use is the math that is taught in ALGEBRA 1! 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!

The Objective of this presentation is show how: to simplify algebraic expressions by using the rules for order operations to evaluate algebraic expressions.

) ( ) ( ) ( ) ( Two Examples 14- 10 3 +10 = 14 -30+10 = -16 +10 = -4 = Simplifying Expressions Rules Perform operations within parenthesis first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Two Examples (a) 14- 10 3 +10 = 14 -30+10 = -16 +10 = -4 (b) = ) ( - 9 1 4 10 ) ( - 0.11 0.25 10 1 ) ( 0.14 0.1 ) ( 0.14 10 1 = 0.014 = =

14 - +10 = (10 3) = (10 3) 14- +10 = = 14 -30+10 = = -16 +10 = -4 Simplifying Expressions Rules Perform operations within parenthesis first. Perform operations within parenthesis first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Add (subtract) in order from left to right. Two Simple Examples (a) 14 - +10 = (10 3) ? = (10 3) 14- +10 = = 14 -30+10 = = -16 +10 = -4

= ) ( - 9 1 4 10 0.11 0.25 0.14 0.1 0.014 (b) 0.014 Rules used? Perform operations within parenthesis first. Multiply (divide) in order from left to right. Multiply (divide) in order from left to right. Another way that technicians, scientists and engineers often simplify this type of algebraic expression. (9-4) (4 9) ) ( - 9 1 4 10 10 1 ( ) ) ( 5 36 10 1 = = = Rule to use first? 0.014 ) ( 0.14 10 1 = = 0.014 Perform operations within parenthesis

) ( ( ) ) ( = - d 1 b 10 10 1 (d – b) (b d) - 9 1 4 10 Simplifying Expressions Rules Perform operations within parenthesis first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Two Generalization Examples = ) ( - d 1 b 10 10 1 ( (d – b) (b d) ) (a) For the previous problem, b was equal to 4 and d was equal to 9 ) ( - 9 1 4 10

) ( ) ( = - n2 1 n1 10 (n2– n1) (n1 n2) = - d 1 b 10 (d – b) (b d) Technical workers often use different symbol combinations for the letters b and d. = ) ( - n2 1 n1 10 (n2– n1) (n1 n2) (b) This time the symbol n1 replaces the letter b and the symbol n2 replaces the letter d.

Evaluation of a new expression ) ( - n2 1 n1 10 2 = ? This time let n1 equal 2 and n2 equal 3 = ) ( - 3 1 2 10 (9 -4) (4 9) ) ( 5 36 10 1 = ) ( 0.14 10 1 = = ? 0.014 NOTE: The calculations inside the parenthesis were completed before multiplying by one tenth.

[ ] [ ] [ ] [ ] [ ] [ ] Reciprocal Expressions 10 10 1 = = 0.10 Rules Perform operations within parenthesis first. Perform operations within parenthesis first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Three easy examples of reciprocal expression manipulations a) -1 10 [ ] 10 [ ] 1 = = 0.10 There is nothing to do inside this parenthesis b) 2 + 3(2) +2 [ ] 1 = 2 + 6 +2 [ ] 1 = 10 [ ] 1 = -1 10 [ ] = 0.10 There is something to do inside this parenthesis

[ ] [ ] [ ] [ ] [ ] [ ] ) ( ) ( ) ( ) ( Reciprocal Expressions 10 1 10 Rules Perform operations within parenthesis first. 10 [ ] 1 -1 10 [ ] ) ( - 4 1 3 20 [ ] = [ ] 1 ) ( 4 2 20 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. c) = = A typical reciprocal (inverse) expression used in technology ) ( - n2 1 n1 2 10 [ ] -1 ) ( - n2 1 n1 2 10 [ ] = These two expressions are same.

[ ] ) ( Reciprocal Expressions - n2 1 n1 10 = Practice Problem Rules Perform operations within parenthesis first. Practice Problem What is the value of this expression when n1 equals 2 and n2 equals 3? ) ( - n2 1 n1 2 10 [ ] = -1

[ ] [ ] [ ] [ ] [ ] ) ( ) ( ) ( ) ( - n2 1 n1 10 = - 3 1 2 10 (9 -4) Perform operations within all parenthesis first! ) ( - n2 1 n1 2 10 [ ] -1 NOTE: 2 = 2 times 2 = 4 3 2 n1 equals 2 and n2 equals 3 = 3 times 3 = 9 = ) ( - 3 1 2 10 [ ] -1 (9 -4) (4 9) = ) ( 5 36 10 1 [ ] -1 ) ( 0.14 10 1 [ ] -1 = 0.014 [ ] -1 The calculation of the inverse is the last thing done. = 71.4 =

[ ] [ ] [ ] [ ] [ ] ) ( ) ( ) ( ( ) ( ) 0.014 = = - 3 1 2 10 (9 -4) Perform operations within parenthesis first NOTE: 0.014 [ ] -1 1 0.014 ( ) = 1) 1 0.014 ( ) is the inverse of the number 71.4 2) = ) ( - 3 1 2 10 [ ] -1 (9 -4) (4 9) = ) ( 5 36 10 1 [ ] -1 ) ( 0.14 10 1 [ ] -1 = 0.014 [ ] -1 The calculation of the inverse is the last thing done. = 71.4 =

) ( ) ( ( ) ( ) - n2 1 n1 (n2– n1) (n1 n2) 3 quick review questions to see what we remember 1) What are, in the correct order of use, the rules for simplifying algebraic expressions? Perform operations within parenthesis first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. 2) What is another way to write the following algebraic expression? ) ( - n2 1 n1 ) ( (n2– n1) (n1 n2) = 3) What is (a) the inverse of ? 1 0.014 ( ) 71.4 1 0.014 ( ) (b) the reciprocal of the the number 71.4?

[ ] [ ] ) ( ) ( - n2 1 n1 10 - n2 1 n1 10 What do you think? 1) Are the two algebraic expressions show below equal? Why/why not? (a) (b) ) ( - n2 1 n1 2 10 [ ] -2 ) ( - n2 1 n1 2 10 [ ] 2) Is the inverse of a number always the same as the reciprocal of that number? Why/Why not?