Chapter 1 Elementary Arithmetic

Section 1.4 Powers and Roots

1.4.2 Calculating using Powers


The following calculation rules allow one to transform and simplify expressions containing powers or roots:
Info 1.4.13
 
For a,b,a,b>0,p,q, the following exponent rules hold:

ap · bp =(a·b )p , ap bp = ( a b )p , ap · aq = ap+q , ap aq = ap-q ,( ap )q = ap·q .


Note that, generally, ( ap )q a pq , i.e. multiple powers should be bracketed. For example, ( 23 )2 = 82 =64, but 2( 32 ) = 29 =512.  
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Example 1.4.14
Without brackets one reads a pq as a( pq ) , i.e. for example,

2 34 = 23·3·3·3   =   281   =  2417851639229258349412352     (exponent evaluated first) ( 23 )4 = 84   =  4096     (bracket evaluated first) .

Alternatively, we could have used the exponent rules to calculate ( 23 )4 = 2(3·4) = 212 =4096.

Exercise 1.4.15
The following expressions can be simplified using the exponent rules:
  1. 33 · 35 · 3-1 = .

  2. 42 · 32 = .


However, when comparing powers and roots, one should take care: not only the values, but also the signs of exponent and base control whether the value of the power is large or small:
Example 1.4.16
For a positive base and a negative exponent, the value of the power decreases when increasing the base:

2-1 = 1 2   =  0.5 3-1 = 1 3   =  0. 3 4-1 = 1 4   =  0.25   etc.


For a negative base the sign of the power alternates when increasing the exponent:

(-2 )1 = -2 (-2 )2 = 4 (-2 )3 = -8 (-2 )4 = 16   etc.


Extracting the root (or exponentiation with a positive number smaller than one) decreases a base >1, but increases a base <1:

2 = 1.414    <    2 3 = 1.732    <    3 0.5 = 0.707    >    0.5 0. 3 = 0.577    >    0. 3    etc.



Exercise 1.4.17
Arrange the following powers in order of size considering the signs of bases and exponents: 23 , 2-3 , 32 , (-3 )2 , (-3 )-2 , 3 1 2 , 2 1 3 :  

  < 
  < 
  < 
  < 
  < 
  = 
 .