## 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.

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.

In this link, you will find not only the solutions but also the explanations.

http://www.mathgoodies.com/lessons/vol5/challenge_vol5.html

These problems have a higher level. They can have hints but no explanation.

## Lesson 1: Integers (I)

September 22, 2010

Vocabulary on Integers

” Lesson 1: Integers” (Notes)

Word Problems on Integers

I Have to Know by the End of this Lesson

### 1. Revising concepts

Last year you learnt the essential notions about integers but you probably need some help to remember. Here you are some links on the main concepts

Integers on number lines

Absolute value and opposite integers

Compare and order integers

Integer inequalities with absolute values

Here you are a funny video “Integers Rap”. Coluld you write a comment giving the translation of the lyrics into Spanish?

### 2. Operations with integers

To add up two integers we do the following:

• If both of them have the same sign, we add up their absolute values and we write their sign
• If they have different sign, we subtract their absolute values and we write by the result the sign of the integer that has the bigger absolute value.

Subtracting a number is adding up its opposite. We add up the minuend to the opposite of the subtrahend.

 Rule Example Two like signs become a positive sign +(+) (+3)+(+2) = 3 + 2 = 5 -(-) (+6)-(-3) = 6 + 3 = 9 Two unlike signs become a negative sign +(-) (+7)+(-2) = 7 – 2 = 5 -(+) (+8)-(+2) = 8 – 2 = 6

Here you are a video on subtracting integers

The following  exercises are elemental. Click on the following links if you want to refresh your knowledge:

All subtraction

MULTIPLICATION AND DIVISION

To work out the product, or quotient, of two integers:

1st Multiply, or divide,  their absolute values

2nd Write + sign if they have like signs or write – sign if they have unlike signs.

Now remember the rule of the signs

When You Multiply …

 Example × two positives you get a positive: (+3 ) · (+2) = +6 × a positive and a negative you get a negative: (-3) · (+ 2) = -6 × a negative and a positive you get a negative: (+3) · (-2) = -6 × two negatives you get a positive: (-3) · (-2) = +6

The same rules are valid for division

 Example : two positives you get a positive: (+12 ) : (+2) = +6 : a positive and a negative you get a negative: (-12) : (+ 2) = -6 : a negative and a positive you get a negative: (+12) : (-2) = -6 : two negatives you get a positive: (-12) : (-2) = +6

Practise multiplication and division on these links

Multiplying signed integers

Dividing signed integers

Practise  these operations with numbers in brackets on the following links: