Exponents, Parentheses, and the Order of Operations (OOOs) Section 1.9 (68)

Objectives Learn the meaning of exponents Evaluate operations containing exponents Learn the difference between –x 2 and (-x) 2 Learn the Order of Operations (OOOs) Learn the use of parentheses Evaluate expressions containing variables

1.9.1 Learn the Meaning of Exponents (68) If there is a method to write less, you can bet a mathematician will find a way. Rather than writing (3 times itself five times), they will write 3 5 (3 to the fifth power). In this case, the 3 is called the base, the 5 is called the exponent. Note that every integer can be factored into a product of primes (each possibly to a power)

1.9.2 Evaluating Expressions Containing Exponents (69) Example: 2 4 = 2222 = 16 Later will we determine that if we have there is a negative value in front of a value to a power, the power is done first. Examples: (3) 3 = 333 = 27 (-2) 4 = (- 2)(-2)(-2)(-2) = = = -64 It is important that you understand the difference between the two above. (2/3) 2 = (2/3)(2/3) = 4/9

Simplification By default any value to the 1 st power is itself and is normally not written. x 1 ≡ x Similarly, when we use variables, you seldom see 1x instead of just plain x. Note we also assume there is a between a number and a letter of between letters. Examples: xxxyy=35x 3 y 2 5x5y 2 = 5 2 xy 2

Hint Note the difference between: x + x + x + x + x = 5x x x x x x = x 5 Examples: = 4 3 = = 2 3 = 8

1.9.3 Learn the Difference Between -x 2 and (-x) 2 (69) If there is no parentheses around a value to a power, the power is done first. -x 2 ≡ -1●x 2 Try and remember that you do powers before subtraction. Examples: -3 2 = -1●3●3 = -9 (-2) 4 = (-2)(-2)(-2)(-2) = -16

1.9.4 Order of Operations (OOOs) (71) Forget “My dear Aunt Sally”. Sometimes you will divide before you multiply, sometimes you will subtract before you add. The Order of Operations is: Parentheses (inner ones first) ( ), [ ], { } Powers (exponents) Multiply/Divide LEFT TO RIGHT Add/Subtract LEFT TO RIGHT

Examples ● 4 2 – ● 16 –

Examples [ -3 + (100 ÷ 5 2 )] [ -3 + (100 ÷ 25)] [ ] [ 1 ]

Examples (14 ÷ 2) + 5(3 – 2) (3 – 2) ( 1 )

Examples -9 – 72 ÷ – 72 ÷

Examples ÷ ÷ (-5) ÷ ÷

Examples ( 4/7 ) – ( 3/5 )( 2/9 ) 2 1 ( 4/7 ) – [(3)(2)]/[(5)(9)] ( 4/7 ) – [(1)( 2)]/[(5)(3)] ( 4/7 ) – ( 2/15 ) LCD 105 (4/7)(15/15) – (2/15)(7/7) [60 – 14]/105 46/105

Examples { [ ( 9 6 ) + 7 ] - 12 } ÷ { [ ( 54 ) + 7 ] – 12 } ÷ 5 { } ÷ 5 { 49 } ÷ 5 49/5

1.9.6 Evaluating Expressions Containing Variables (74) To evaluate an expression for a specific value of a variable(s), first put the variable in parentheses, then evaluate the expression. Examples: 3x + 2 for x = 2 3(x) + 2 3(2)

Examples Evaluate x 2 for x = 9 (x) 2 ; (9) 2 ; 81 -x 2 for x = 9 -(x) 2 ; -(9) 2 ; -81 x 2 for x = -2 (x) 2 ; (-2) 2 ; 4 -x 2 for x = 9 -(x) 2 ; -(-2) 2 ; -4

Hint Note the difference between -5 2 and (-5) 2 This is a very important concept.

Example Sometimes the variable is to a power. That is why we put the variable in parentheses. Evaluate: 2x 2 – 3x + 4 for x = ( 2/3 ) 2(x) 2 - 3(x) + 4 2( 2/3 ) 2 - 3( 2/3) + 4 2( 4/9 ) - 6/3 + 4 ( 8/9 ) – ( 18/9 ) + ( 36/9 ) - ( 10/9 ) + ( 36/9 ) ( +26/9 ) or 2 8/9

Example Sometimes we have more than one variable Evaluate: -y 2 + 2(x + 7) – 3 for x = -2, y = -1 -(y) 2 + 2([x] + 7) – 3 -(-1) 2 + 2([-2] + 7) – 3 -(-1) 2 + 2( +5 ) - 3 -( 1 ) + 2( +5 ) – (+10)

Objectives Learn the meaning of exponents Evaluate operations containing exponents Learn the difference between –x 2 and (-x) 2 Learn the Order of Operations (OOOs) Learn the use of parentheses Evaluate expressions containing variables

Exponents, Parentheses, and the Order of Operations (OOOs) Section 1.9 (68)