Foundation of Economic Analysis

Foundation of Economic Analysis - Transcript

Foundations of Economic Analysis Topic Quantitative Techniques Lecture 3
Homagni Choudhury Economic Studies University of Dundee

Schedule Office Hours
20th Jan Tue Linear Functions Straight Lines 22nd Jan Thu Solving simultaneous equations 27th Jan Tue Indices 2nd Feb Mon Non Linear functions 19th Feb Thu Class TEST Office hours Thursdays 1 30 2 30 pm Room G 4 Economic Studies Building e mail h choudhury dundee ac uk Phone 01382 385166

Note Only for next week
Office hours is from 2 30 to 3 30 PM on Wed 4th Feb Normal Office hours otherwise Thursdays 1 30 2 30 PM

Today s topic Indices
Squares cubes and higher powers in arithmetic Square roots and cube roots in arithmetic Powers and roots in algebra The 8 Rules

The base and the power
The product 2x2x2 may be written as 23 and read as 2 to the power of 3 2 is called the base and 3 is called the exponent or index The exponent tells you how many times the base is to be multiplied by itself

Examples
24 2x2x2x2 16 23 2x2x2 8 22 2x2 4 21 2

When the exponent is 1 it is not usually shown explicitly for eg We write 4 and not 41

Powers of positive and negative numbers
When a positive number is raised to any power the result is always positive Examples
52 5x5 25 33 3x3x3 27 105 10x10x10x10x10 100 000

Powers of positive and negative numbers
When a negative number is raised to a power the result may be either positive or negative
When it is raised to an even power the result is positive When it is raised to an odd power the result is negative

Powers of positive and negative numbers
Examples
2 2 2 x 2 4 2 3 2 x 2 x 2 8 2 4 2 x 2 x 2 x 2 16 2 5 2 x 2 x 2 x 2 x 2 32 2 6 2 x 2 x 2 x 2 x 2 x 2 64

Powers of fractions
Fractions can also be raised to a power
111 1 1 2228 2 2 2 2 4 3 3 3 9 8 2 2 2 2 27 3 3 3 3
3 2 3

Square roots and cube roots in arithmetic
Square root is the reverse of square 22 2x2 4 2 2 2 x 2 4 Therefore square root of 4 is 2 or 2 i e 4 2 or 2 Negative numbers have no real square root

Square roots and cube roots in arithmetic
Cube root is the reverse of cube 23 2x2x2 8 2 3 2 x 2 x 2 8 Therefore cube root of 8 is 2 and cube root of 8 is 2 i e 3 8 2 and 3 8 2 Negative numbers do have real cube roots A similar pattern is followed for higher roots

Powers and roots in algebra
Algebraic terms can also contain squares cubes and higher powers X3 X x X x X a2 a x a The x and a are again called base while the superscript is called the exponent or index

Powers and roots in algebra
Square roots
a x a a2 a x a a2 So square root of a2 is a or a i e a2 a or a

Cube roots
a x a x a a3 a x a x a a 3 So cube root of a3 is a and a 3 is a i e 3 a3 a and 3 a 3 a

The Eight Rules Definitions
You need to know eight rules definitions in order to be able to perform algebra when powers are involved The first four The Index Rules The last four The Index Definitions

Index Rule 1 Multiplication with powers
Consider a2 x a3 a2 a x a a3 a x a x a Thus a2 x a3 a x a x a x a x a But we know a x a x a x a x a a5 So a2 x a3 a5 Since 5 2 3 a5 may be written as a2 3 Therefore a2 x a3 a2 3 a5 This can be generalised to give the first rule

Index Rule 1 Multiplication with powers am x an am n
When two numbers with the same base are multiplied together we add the indices This rule only applies to numbers with same base Examples
23 x 26 23 6 29 a2 x a5 x a3 a2 5 3 a10

Index Rule 2 Division with powers
a6 Consider 2 a a6 a a a a a a a2 a a a6 a a a a a a Thus 2 a a a a a4 a a a Since 6 2 4 a 4 can be written as a 6 2 a6 So 2 a 6 2 a 4 a This can be generalised to give the second rule

Index Rule 2 Division with powers

a m n a n a
When one number is divided by another number with the same base we subtract the indices This rule only applies to numbers with same base

m

Index Rule 3 Raising a power to a power
Consider a2 3 This is the cube of a2 Therefore a2 3 a2 x a2 x a2 a6 Since 2x3 6 a6 can be written as a2x3 So a2 3 a2x3 a6 This can be generalised to give the third rule

Index Rule 3 Raising a power to a power

am n amxn
Examples
34 3 34x3 312 531441 x2 7 x2x7 x14

Index Rule 4 Removing brackets with powers

Consider ab 3 ab 3 ab x ab x ab axbxaxbxaxb axaxaxbxbxb a3xb3 a3b3 This can be generalised to give the fourth rule

Index Rule 4 Removing brackets with powers

ab n anbn
Examples
4x3 2 42 x 32 16 x 9 144 xy 6 x6 y6

Index Definition 1 Meaning of the power 0 Consider am am Any number divided by itself equal 1 so am am 1 Also from IR 2 am am am m a0 Combining these two results we have a0 1 which is the first definition

Index Definition 1 Meaning of the power 0

a0 1
Any number or algebraic term raised to the power 0 is equal to 1 Examples
30 1 x0 1 3 4 0 1 5x 0 1

Index Definition 2 Meaning of a negative index
1 Consider n a From ID1 a 0 1 Therefore we can write 1 a0 n a 0 n using IR2 an a 1 n So n a which is our second definition a

Index Definition 2 Meaning of a negative index

a n 1 an
Therefore if an is multiplied by a n then we get 1 an a n an n an n a0 1 Or an a n an 1 an 1

Index Definition 3 Meaning of a fractional index
Consider a 2 a From IR3 a1 2 2 a Combining the results a 2 a1 2 2 a So a a1 2 Similarly we can show 3 a a1 3 4 a a1 4 and so on which gives the third definition

Index Definition 3 Meaning of a fractional index a1 n n a
Although all positive numbers have two square roots one positive and one negative the rule is that we take a1 2 to be the positive square root only and similarly for other fractional indices

Index Definition 4 Fractional Powers the general case
Consider a
2 3 2 3 2 1 3 1 23

Using IR3 a a
1 23

a

Using ID3 a 3 a 2 Combining these we have a
2 3 3

a2
2 3

In a similar way we can show that a 3 a 2 and hence the fourth definition

Index Definition 4 Fractional Powers the general case

a a
n m
Examples 4 3 2 43 64 8 etc

m n

a
n

m

The Basic Index Rules
am x an am n

a m n a n a a
am n amxn ab n anbn

m

The Basic Index Definitions
a0 1 a n 1 an a1 n n a

m nm n m an a a



Some Examples

End of lecture
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