BASIC ALGEBRAIC OPERATIONS

1
UNIT 12

• BASIC ALGEBRAIC OPERATIONS

2
ADDITION

• Only like terms can be added. The addition of
  unlike terms can only be indicated
  
• Procedure for adding like terms
• Add the numerical coefficients, applying the
  procedure for addition of signed numbers
  
• Leave the literal factors unchanged

a. 6y (5y) 1y y Ans
b. 13ab (11ab) 24ab Ans
3
ADDITION

• Procedure for adding expressions that consist of
  two or more terms
  
• Group like terms in the same column
• Add like terms and indicate the addition of the
  unlike terms
  
• Add 5y (3x) 6x2y and (4x) (2y)
  (2x2y)
  
• Group like terms in the same column
• Add the like terms and indicate the
  addition of
  the unlike terms

4
SUBTRACTION

• Just as in addition, only like terms can be
  subtracted
  
• Each term of the subtrahend is subtracted
  following the procedure for subtraction of signed
  numbers
  
• Subtract (7x2 7xy 15y2) (8x2 5xy
  10y2)Change the sign of each term in the
  subtrahend and follow the procedure for
  addition of signed numbers

5
MULTIPLICATION

• In multiplication, the exponents of the literal
  factors do not have to be the same to multiply
  the values
  
• Procedure for multiplying two or more terms
• Multiply the numerical coefficients, following
  the procedure for multiplication of signed
  numbers
  
• Add the exponents of the same literal factors
• Show the product as a combination of all
  numerical and literal factors
  
• Multiply (4)(5x)(6x2y)(7xy)(2y3)
• Multiply all coefficients and add exponents of
  the same literal factors
  

(4)(5)(6)(7)(2)(x1 2 1)(y1 1 3)
1680x4y5 Ans
6
MULTIPLICATION

• Procedure for multiplying expressions that
  consist of more than one term within an
  expression
  
• Multiply each term of one expression by each term
  of the other expression
  
• Combine like terms

b. (2a 3b)(5a 2b)
a. 2x(3x2 2x 5)
(2a)(5a) (2a)(2b) (3b)(5a) (3b)(2b)
2x(3x2) 2x(2x) (2x)(5)
6x3 4x2 10x Ans
10a2 4ab 15ab 6b2
10a2 11ab 6b2 Ans
7
DIVISION

• Procedure for dividing two terms
• Divide the numerical coefficients following the
  procedure for division of signed numbers
  
• Subtract the exponents of the literal factors of
  the divisor from the exponents of the same letter
  factors of the dividend
  
• Combine numerical and literal factors

8
DIVISION

• Divide (40a3b4c5) ? (4ab2c3)

(10)(a3 1)(b4 2)(c5 3)
10a2b2c2 Ans
9
DIVISION

• Procedure for dividing when the dividend consists
  of more than one term
  
• Divide each term of the dividend by the divisor,
  following the procedure for division of signed
  numbers
  
• Combine terms
• Divide

10
POWERS

• Procedure for raising a single term to a power
• Raise the numerical coefficients to the indicated
  power following the procedure for powers of
  signed numbers
  
• Multiply each of the literal factor exponents by
  the exponent of the power to which it is raised
  
• Combine numerical and literal factors
• Procedure for raising two or more terms to a
  power
  
• Apply the procedure for multiplying expressions
  that consist of more than one term

(2x2y3)2 22(x2)2(y3)2 4x4y6 Ans
(x y)2 (x y)(x y) x2 xy xy y2
x2 2xy y2 Ans
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ROOTS

• Procedures for extracting the root of a term
• Determine the root of the numerical coefficient
  following the procedure for roots of signed
  numbers
  
• The roots of the literal factors are determined
  by dividing the exponent of each literal factor
  by the index of the root
  
• Combine the numerical and literal factors
• Solve

2a2b3c Ans
12
REMOVAL OF PARENTHESES

• Procedure for removal of parentheses preceded by
  a plus sign
  
• Remove the parentheses without changing the signs
  of any terms within the parentheses
  
• Combine like terms
• 7x (4x 3y 2) 7x 4x 3y 2 11x
  3y 2 Ans
  
• Procedure for removal of parentheses preceded by
  a minus sign
  
• Remove the parentheses and change the sign of
  each term within the parentheses
  
• Combine like terms
• 9a (4a 2b 6) 9a 4a 2b 6 13a 2b
  6 Ans

13
COMBINED OPERATIONS

• Expressions that consist of two or more different
  operations are solved by applying the proper
  order of operations
  
• Simplify 15x 4(2x) x

15x 8x x
23x x 24x Ans

• Simplify 3x x (x2y3)22

3x x x4y62
2x x4y62
(2x x4y6)(2x x4y6)
4x2 2x5y6 2x5y6 x8y12
4x2 4x5y6 x8y12 Ans
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SCIENTIFIC NOTATION

• In scientific notation, a number is written as a
  whole number or decimal between 1 and 10
  multiplied by 10 with a suitable exponent
  
• 1,750,000 is written as 1.75 106 in scientific
  notation
  
• 0.00065 is written as 6.5 104 in scientific
  notation
  
• 9.8 103 in scientific notation is written as
  9,800 as a whole number
  
• The problem below uses scientific notation when
  multiplying two numbers
  
• (1.2 103)(5 101) (1.2)(5) (103)(101)
  6 102 Ans

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METRIC PREFIXES

• The table below lists metric prefixes applied to
  very large and very small numbersPrefix
  Symbol Meaning Factor Value
  Power of 10tera T one
  trillion 1,000,000,000,000
  1012giga G one billion
  1,000,000,000 109 mega
  M one million 1,000,000
  106kilo k one
  thousand 1,000
  103------ --- BASE UNIT
  1 100milli
  m one thousandth 0.001
  10-3micro µ
  one millionth 0.000001
  10-6nano n one billionth
  0.000000001 10-9

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METRIC CONVERSIONS

• The base unit listed on the previous slide can
  stand for any number of things, including meters
  (m), amperes (A), watts (W), and volts (V)
  
• We will use the unity fraction method learned
  earlier to convert from one metric unit to
  another
  
• Express 12 volts as millivolts

17
BINARY NUMERATION SYSTEM

• The binary number system uses only the two digits
  0 and 1. These two digits are the building blocks
  for the binary code that is used to represent
  data and program instructions for computers
• Place values for binary numbers are shown below

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EXPRESSING BINARY NUMBERS AS DECIMAL NUMBERS

• We use a subscript 2 to show that a number is
  binary
  
• Express the binary number 1012 as a base 10
  decimal number
  
• 1012 1(22) 0(21) 1(20) 4 0 1 510
  Ans
  
• Express the decimal number 1910 as a binary
  number
  
• The largest power of two that will divide into 19
  is 24 or 16
  
• 19 1(24) 0(23) 0(22) 1(21) 1(20)
  100112 Ans

19
PRACTICE PROBLEMS

• Perform the indicated operations and simplify1.
  6a 7a 9b2. (3xy) 4x (5xy2) 5xy
  (7x)3. 3.07ab 7.69c (5.76ab) 9d
  (11.2c)4. 1/2x (2/3y) 1/4z (1/3z)
  2/3x5. 4x2y (5xy2) 7xy2 (2x2y)6.
  7a 3a7. 10x (20x)8. (3y2 4z)
  (2y2 4z)9. 1 1/2ab 1 2/3ab10.
  (2.04t2 7.6t 7) (3t2 6.7t 4)

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PRACTICE PROBLEMS
11. (3ab)(4a2b2) 12. (1/2x)(1/3y2)(1/4x3) 1
3. (a b)(a b) 14. (2x2 3y)(3x2 2y) 15
. 16y2 ? 4y 16. 1 1/3 a2b3 ? 2/3ab 17. (2.4x
3y3 4.8x2y2 24x) ? 1.2x 18. (x2y)3 19. (
2.1a2b3)2 20. (2/3 x3y3z2)3 21. (2m2n p3)2
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PRACTICE PROBLEMS

• 22.
• 23.
• 24.
• 25. 2x (x 2y)
• 26. (x y z) (x2 y2 z)
• 27. 15 (ab a2b b) 4 (ab b)
• 28. (18a4b2) ? (3a2b) b3(b2)
• 29. 5(2x y)2 (x2 y2)
• 30. Express 0.00089 in scientific notation

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PRACTICE PROBLEMS

• 31. Express 4.3 105 in decimal
  (standard) form
  
• Express the answer in scientific notation with
  three significant digits
  
• 33. Express 128 milliamperes as amperes
• 34. Express 260000 microwatts as kilowatts
• 35. Express 1112 as a decimal number
• 36. Express 101012 as a decimal number
• 37. Express 9 as a binary number
• 38. Express 26 as a binary number

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PROBLEM ANSWER KEY

• 1. 13a 9b
• 2. 2xy (3x) (5xy2)
• 3. 8.83ab (3.51c) 9d
• 4. 7/6x (2/3y) (1/12z)
• 5. 2x2y 2xy2
• 6. 4a
• 7. 10x
• 8. 5y2 8z
• 9. 3 1/6ab
• 10. 5.04t2 14.3t 3

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PROBLEM ANSWER KEY

• 11. 12a3b3
• 12. 1/24x4y2
• 13. a2 2ab b2
• 14. 6x4 13x2y 6y2
• 15. 4y
• 16. 2ab2
• 17. 2x2y3 4xy2 20
• 18. x6y3
• 19. 4.41a4b6
• 20. 8/27x9y9z6
• 21. 4m4n2 4m2np3 p6

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PROBLEM ANSWER KEY

• 22. 11a3b2c
• 23. 4x4y3
• 24. 2/3m3n4
• 25. x 2y
• 26. x y x2 y2
• 27. 11 a2b
• 28. 6a2b b5
• 29. 19x2 20xy 6y2
• 30. 8.9 104

• 31. 430,000
• 32. 1.47 103
• 33. 0.128 A
• 34. 0.00026
• 35. 7
• 36. 21
• 37. 10012
• 38. 110102