Lesson 2
The Arithmetic of Powers
Multiplication by addition, and division by subtraction
Page 60
We can write out
27 and
24 in full to show again why the number of times the base (2 in this case) occurs as a factor in the quotient is the difference between the number of times it occurs as a factor in the dividend (
27 ) and the number of times in the divisor (
24 ):
( 128 )/( 16 ) = ( 27 )/( 24 ) = ( 2 × 2 × 2 × 2 × 2 × 2 × 2 )/( 2 × 2 × 2 × 2 ) = 2 × 2 × 2 = 8.
The cancellation of 2’s in the dividend and divisor leaves three 2’s to be used as factors in forming the quotient.
Here are four sets of operations involving exponents and powers of numbers. Pick out the group containing a mistake.
( 47 )/( 42 ) = 45
( b m )/( b n ) = b ( m − n )
( 74 )/( 72 ) = 49
( 39 )/( 33 ) = 33
⎫ ⎬ ⎭ This group contains a mistake.
( b 16 )/( b n ) = b ( 16 − n )
( 53 )/( 51 ) = 25
( 104 )/( 102 ) = 100
( b 2 )/( b ) = b 1
⎫ ⎬ ⎭ This group contains a mistake.
( 45 )/( 42 ) = 64
( 6m )/( 63 ) = 6( m − 3)
( 43 )/( 42 ) = 4
( 122 )/( 12 ) = 12
⎫ ⎬ ⎭ This group contains a mistake.
( ( 82 ) ( 84 ) )/( 83 ) = ( 86 )/( 83 ) = 83
( b 10 )/( b 7 ) = b 3
( 33 )/( 31 ) = 9
( 63 )/( 6 ) = 36
⎫ ⎬ ⎭ This group contains a mistake.
Answer to Self-Test Question 15, Lesson 2 :
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