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

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Page 1: 1 Simplifying Expressions Simplifying Expressions in Algebraic Expressions Applications in Atomic Sciences

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Simplifying Expressions

Simplifying Expressions in Algebraic ExpressionsApplications in Atomic Sciences

Page 2: 1 Simplifying Expressions Simplifying Expressions in Algebraic Expressions Applications in Atomic Sciences

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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!

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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.

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Simplifying Expressions

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 =

=)( - 91

41

101

=

)( - 0.110.25101

)( 0.14101 = )( 0.140.1 = 0.014

-16 +10

= -414- +10

10 3

(a)

(b)

Rules

=

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Simplifying Expressions

Simplifying Expressions

Perform operations within parenthesis first.

Add (subtract) in order from left to right.

Two Simple Examples

(10 3)14- +10

=

14 -30+10 =-16 +10

= -4

14 - +10

=(10 3) (a) ?

=

=

=

Multiply (divide) in order from left to right.

Perform operations within parentheses first.

Add (subtract) in order from left to right.

Rules

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Simplifying Expressions

=)( -91

41

101

=

)( - 0.110.25101

)(0.14101

= )(0.140.1 = 0.014

Another way that technicians, scientists and engineers often simplify this type of algebraic expression.

=)( - 91

41

101 )(10

1 (9-4)

(4 9)= )( 5

36101 =

0.014)(0.14101 =

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.

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Simplifying Expressions

Two Generalization Examples

=)( -d1

b1

101 )(

10

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)

)( - 91

41

101

For the previous problem, b was equal to 4 andd was equal to 9

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Simplifying Expressions

=)( -n2

1 n1

1101 )( (n2–

n1)10

1

(n1 n2)

This time the symbol n1 replaces the letter b and the symbol n2 replaces the letter d.

(b)

=)( -d1

b1

101 )( (d – b)

101

(b d)

Technical workers often use different symbol combinations for the letters b and d.

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Simplifying Expressions

Evaluation of a new expression

)( -n2

1 n1

1

10

12 2

This time let n1 equal 2 and n2 equal 3

= ? 0.014)(0.14101 =

=)( -3

1 2

1101

)( (9 -4)

101

(4 9)2 2

= ?

NOTE: The calculations inside the parentheses were completed before multiplying by one tenth.

)( 5 3610

1 =

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Simplifying Expressions

Perform operations within parenthesis first.

Reciprocal Expressions

10[ ]1

=-1

10[ ] = 0.10

-1

10[ ] =0.10

Three easy examples of reciprocal expression manipulationsa)

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

=

RulesPerform operations within parentheses first.

Add (subtract) in order from left to right.

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Simplifying Expressions

These two expressions are same.

)( -n2

1 n1

112 210[ ]

1=

-1

)( -n2

1 n1

112 210[ ]

c)

)( - 41

4320[ ]1

= =

[ ]1

)( 4220

-1

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.

RulesPerform operations within parentheses first.

Reciprocal Expressions

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Simplifying Expressions

What is the value of this expression when n1 equals 2 and n2 equals 3?

)( -n2

1 n1

112 210[ ]

1=

-1

)( -n2

1 n1

112 210[ ]

Practice Problem

RulesPerform operations within parentheses first.

Reciprocal Expressions

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Simplifying Expressions

NOTE:

22

= 2 times 2 = 4

32= 3 times 3 = 9

= =

Perform operations within all parentheses first!

=)( -3

1 2

1

101

2 2[ ]-1

)(101 (9 -

4)(4 9)

[ ]-1

== )( 5 3610

1[ ]-1

)(0.14101[ ]

-1

0.014[ ]-1

71.4

n1 equals 2 and n2 equals 3

The calculation of the inverse is the last thing done.

)( -n2

1 n1

112 210[ ]

-1

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Simplifying Expressions

= =

Perform operations within parentheses first

=)( -3

1 2

1

101

2 2[ ]-1

)(101 (9 -

4)(4 9)

[ ]-1

== )( 5 3610

1[ ]-1

)(0.14101[ ]

-1

0.014[ ]-1

71.4The calculation of the inverse is the last thing done.

1

0.014( )0.014[ ]-1NOTE:

=

1

0.014( ) is the inverse of the number 71.4

1)

2)

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Simplifying Expressions

3 quick review questions to see what we remember

1)What are, in the correct order of use, the rules for simplifyingalgebraic expressions?

2)What is another way to write the following algebraic expression?

71.43)

Perform operations within parentheses first.

Multiply (divide) in order from left to right.

Add (subtract) in order from left to right.

)( -n2

1 n1

1 )( (n2– n1)(n1 n2)

=

What is

(b)

(a) the inverse of ? 1

0.014( )1

0.014( )the reciprocal of the the number 71.4?

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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

)( -n2

1 n1

112 210[ ]

2)

(b)

)( -n2

1 n1

112 210[ ]

1

2

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Simplifying Expressions