1. 1 Simplifying Expressions Simplifying Expressions in Algebraic Expressions Applications in Atomic Sciences

2. 2 Simplifying Expressions 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!

3. 3 Simplifying Expressions The Objective of this presentation is show how: to simplify algebraic expressions by using the rules for order operations to evaluate algebraic expressions.

4. 4 Simplifying Expressions Perform operations within parentheses first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Two Examples 14 -30+10 = = )( - 9 1 4 1 10 1 = )( - 0.110.25 10 1 )( 0.14 10 1 = )( 0.14 0.1 = 0.014 -16 +10= -414-+10103 (a) (b) Rules =

5. 5 Simplifying Expressions Perform operations within parenthesis first. Add (subtract) in order from left to right. Two Simple Examples (103)14-+10= 14 -30+10 = -16 +10= -4 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

6. 6 Simplifying Expressions = )( - 9 1 4 1 10 1 = )( - 0.110.25 10 1 )(0.14 10 1 = )(0.14 0.1 = 0.014 Another way that technicians, scientists and engineers often simplify this type of algebraic expression. = )( - 9 1 4 1 10 1 )( 1 (9-4) (4 9) = )( 5 3610 1 = 0.014 )( 0.14 10 1 = Rules used? Perform operations within parentheses first. Multiply (divide) in order from left to right. (b) = 0.014 Rule to use first? Perform operations within parenthesis 0.014 Multiply (divide) in order from left to right.

7. 7 Simplifying Expressions Two Generalization Examples = )( - d 1 b 1 10 1 )( 1 (d – b) (b d) Simplifying Expressions Perform operations within parentheses first. Multiply (divide) in order from left to right. Add (subtract) in order from left to right. Rules (a) )( - 9 1 4 1 10 1 For the previous problem, b was equal to 4 and d was equal to 9

8. 8 Simplifying Expressions = )( - n2n2 1 n1n1 1 10 1 )( (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 1 10 1 )( (d – b) 10 1 (b d) Technical workers often use different symbol combinations for the letters b and d.

9. 9 Simplifying Expressions Evaluation of a new expression )( - n2n2 1 n1n1 1 10 1 2 2 This time let n 1 equal 2 and n 2 equal 3 = ?0.014 )( 0.14 10 1 = = )( - 3 1 2 1 1 )( (9 -4) 10 1 (4 9) 22 = ? NOTE: The calculations inside the parentheses were completed before multiplying by one tenth. )( 5 3610 1 =

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

11. 11 Simplifying Expressions These two expressions are same. )( - n2n2 1 n1n1 1 1 2 2 10 [] 1 = )( - n2n2 1 n1n1 1 1 2 2 10 [] c) )( - 4 1 4 3 20 [ ] 1 = = [ ] 1 )( 4 2 10 [] 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

12. 12 Simplifying Expressions What is the value of this expression when n 1 equals 2 and n 2 equals 3? )( - n2n2 1 n1n1 1 1 2 2 10 [] 1 = )( - n2n2 1 n1n1 1 1 2 2 10 [] Practice Problem Rules Perform operations within parentheses first. Reciprocal Expressions

13. 13 Simplifying Expressions NOTE: 2 2 = 2 times 2 = 4 3 2 = 3 times 3 = 9 = = Perform operations within all parentheses first! = )( - 3 1 2 1 10 1 22 [ ] )( 10 1 (9 -4) (4 9) [] = = )( 5 3610 1 [ ] )( 0.14 10 1 [] 0.014 [] 71.4 n 1 equals 2 and n 2 equals 3 The calculation of the inverse is the last thing done. )( - n2n2 1 n1n1 1 1 2 2 10 []

14. 14 Simplifying Expressions = = Perform operations within parentheses first = )( - 3 1 2 1 10 1 22 [ ] )( 10 1 (9 -4) (4 9) [] = = )( 5 3610 1 [ ] )( 0.14 10 1 [] 0.014 [] 71.4 The calculation of the inverse is the last thing done. 1 0.014 () [] NOTE: = 1 0.014 () is the inverse of the number 71.4 1) 2)

15. 15 Simplifying Expressions 3 quick review questions to see what we remember 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? 71.4 3) 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 0.014 () 1 () the reciprocal of the the number 71.4?

16. 16 Simplifying Expressions 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 n1n1 1 1 2 2 10 [] 2) (b) )( - n2n2 1 n1n1 1 1 2 2 10 [] 1 2

17. 17 Simplifying Expressions