Lesson 1: Integers II

September 28, 2010


4. Powers of integers

If  a  is an integer, the product of a  for n  times is denoted by .

an = a · a · …n times….a

 is called the base, the repeated factor,

  is called the index, the times a is repeated

an  and  itself is called the nth power of  a.

4.1. Sign of the power of an integer

 This is a difficult point for you.

If the exponent is a natural number:

  • If the base is positive, the power is positive.
  • If the base is negative, the power is positive if the exponent is an even number and negative if the exponent is an odd number.

4.2. Properties of Powers

 These are the properties of powers. You must learn them by heart.

1.  a0 = 1                                      

2.  a1 = a

3.  an · am = an+m                                 

4.  an:· am =an-m

6. (an)m = an·m

7.  (a : b)n= an: bn                         

8.  (a · b)n= an·  bn

You can practice in


 but the bases are natural numbers, and on the web


 On this web you have to click on ABCD. Some of the questions are not avalaible if you don’t pay (IT IS NOT NECESSARY !!!)


5. Combined operations

 Combined operations are probably the more complicate point on calculations. Remember the order of operations. When you are computing do: 

  1. What is inside the brackets and square brackets.
  2. Powers and roots.
  3. Multiplication and division from left to right.
  4. Addition and subtraction from left to right.

Practise on the links bellow

All types from Difficulty Level 1 Elemental level

All types from Difficulty Level 2 Easy level

All types from Difficulty Level 3 Required level

All types from Difficulty Level 4 Higher level

 Finally here you are a video on combined operations, very useful


6. Word problems with integers

Word problems with integers require a good understanding of the cocept of integer number as well as a comprehesion of the statement. Read it three times at least.

Practise on the bellow links 
In this link, you will find not only the solutions but also the explanations. 
These problems have a higher level. They can have hints but no explanation.


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