Nathan Thomas

Revision Notes: Index Notation: Rules:

Factor Rule: (a x b)m = am x bm Multiplication Rule: am x an = am+n

a-n = 1/an

a0 = 1 Power-on-power Rule: (am) n = amxn Division Rules: am ÷ an = am-n Power ap/q = Root Fractional Indices: a1/2 = √a a2/3 = (3√a)2 = 3√(a)2

This is (Provided m>n). Zero and negative indices: If m=0 then the multiplication rule no longer applies. E.g. a0 = 1. If m= -2 (any negative number) then the multiplication rule doesn’t apply again. E.g. a-n = 1/an. This is known as the negative power rule.

Index Notation

Index Notation and graphs

√

√

Apply the rules!

The multiplication rule:

The division rule:

The power on power rule:

( )

The factor rule:

( )

Graph of y=x1/2

Graph of y=x2/3

Graph of y= x-1

Graph of y= x3

Thorsten Bell

Index Notation

-n-m

=1/(-n)m

x

1/2=rootx

x

0=1

00=undefined

x

-1=1/x

x-n/m

=1/(mrootx)n

-x

2=x

2

x

n/x

m=x

n-m

x

n*x

m=x

n+m

x

n/m=(mrootx)

n

(xn)

m=xn*

m

(x*y)

m=x

m*y

m

If n is a rational number and A>0, the positive solution of

the equation is xn=A is x=A

1/2

(-a)

m=+a

m if m is an even integer or zero

-am

if m is a negative integer

If f(x)=1/xm

, where m is a positive integer, then f’(x)=-m/x

m+1

if f(x)= x

n, where n is a rational number, then f’(x)=nx

n-1

Index Notation notes

- Anything to the power of 0 is 1

- When dividing numbers with powers you subtract the

power - x^3/x^2 = x

- When multiplying numbers with powers you add the

power - 2x^2*2x^2 = 2x^4

- Fraction powers mean you root the number by the

bottom power, and then raise it to the power of the top

number. So 27^2/3 = 9

- negative powers mean that you use the reciprocal and

then raise the denominator by the original power.

So 3^-3 = 1/27

- when multiplying powers in brackets you simply

multiply the powers, so (x^3)^5 = x^15

Ben Smith 12J Index notation

Page 1 of 1

Solve 2-3

=1/23=1/8

Solve 10-4

=1/104 =1/10000

Index Notation

3x-2

= 3

x2

(ab)m= a

m *a

m

(n2)

3=n

6

N0= 1

X1/2

= square root of x

N3 * N

4 =n

7

N5/N

3=N

2

3c2 * 5c

4 = 15c

6

Cameron Parker ^n=To the power of n - R=root

Indices

Cameron Parker ^n=To the power of n - R=root

Indices

With indices, there are a lot of different rules which need to be learnt

There are many different types of indices: Fractions, Integers, positive and negative numbers etc.

Cameron Parker ^n=To the power of n - R=root

Indices-Integers

Positive integers are simple: x^n = x times x, n amount of times e.g 5^2=5x5=25 e.g 5^4=5x5x5x5=625

Negative integers are more complicated:

X^-n = 1/x^n e.g 2^-2 = 1/ 2^n

Anything with x^0 = 1, and anything with x^1 = x

Cameron Parker ^n=To the power of n - R=root

Indices- Fractions

If there is a number, with a fractional power, the rule is very simple, but can have a complicated answer

With fractions, the base is powered by the nominator, and then the dominator becomes the root to the base x^½ becomes Rx^2

e.g 6^¾ = 4R6^3 = 3.834

Cameron Parker ^n=To the power of n - R=root

Indices- Adding, Subtracting, Devising and Multiplying

Multiplying rule also applies for addition:

a^m x a^n = a^m+n e.g 6^3 x 6^4 = 6^7

Dividing rule also applies for subtraction:

a^m / a^n = a^m-n e.g 6^9 / 6^3 = 6^6

Phillip Osborn

Revision Notes -Index Notation

Index notation started as a shorthand way for mathematicians to write multiple amounts of the

letter ‘x’, but was later found out to be so much more than just conventional shorthand and has let

to significant mathematical discoveries.

Simple index notation is used all the time in maths, such as x2 being used all the time as shorthand

for quadratic equations. In general a simple index notation can given the formula:

am = a x a x a x a , with m being the amount of the letter ‘a’.

Here the number ‘a’ is known as the base and the number ‘m’ is known as the index/indices. Note

how at this point although ‘a’ can be of any number ‘m’ must be a positive integer.

There are many rules to simple index notation such as; the multiplication rule, division rule, power-

on-power rule and the factor rule, these are:

The multiplication rule: am x an = am+n

The division rule: am / an = am-n

The power-on-power rule: (am)n = amxn

The factor rule: (a x b)m = am x bm

More complicated indices such as zero or negative indices may seem to make no sense as you

cannot times a number by itself less than 0 times but it is possible by extending the meaning of am. If

you compare the answers of 22 with 2-1 and 22 with 2-2 which are 2, ½, 4, ¼ respectively. You can see

that there appears to be a pattern this can be defined as:

a-m = 1/am

When the index is zero the answers also follow a pattern in which the answer is always 1, This is

formulated as:

a0 = 1

Indices seem to get even more complicated if you look at fraction indices. But if you look at fraction

indices using the power-on-power rule we can easily see a pattern appear, for example if we take

‘m’ as ½ and ‘n’ as 2:

(a1/2)2 = a1/2x2 = a1

Therefore a1/2 would have to be a number whose square is a. There are only two numbers which

equal this; +√a or -√a. So we say a1/2 = √a. This works for other simple fractions and creates the

formula:

a1/m = m√a (note that m is the mth root not just multiplied by m)

For fractional indicies which have a numerator that is greater than 1 we can use a similar formula

which is like the previous one but slightly different, this is:

an/m = (m√a)n = m√an