Download Revision – Indices

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
Index Notation
Power-on-power Rule: (am) n = amxn
Division Rules: am ÷ an = am-n
a
p/q
Power
= Root
1/2
Fractional Indices: a = √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 and graphs
Apply the rules!
The multiplication rule:
The division rule:
The power on power rule:
√
(
The factor rule:
√
-1
Graph of y= x
1/2
Graph of y=x
(
2/3
Graph of y=x
Graph of y= x
3
)
)
Thorsten Bell
Index Notation
-n-m=1/(-n)m
x1/2=rootx
x0=1
00=undefined
x-1=1/x
x-n/m=1/(mrootx)n
-x2=x2
xn/xm=xn-m
xn*xm=xn+m
xn/m=(mrootx)n
(xn)m=xn*m
(x*y)m=xm*ym
If n is a rational number and A>0, the positive solution of
the equation is xn=A is x=A1/2
(-a)m=+am 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/xm+1
if f(x)= xn, 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
(n2)3=n6
3x-2 = 3
x2
Index Notation
N0= 1
3c2 * 5c4 = 15c6
X1/2 = square root of x
(ab)m= am *am
N3 * N4 =n7
Solve 2-3 =1/23=1/8
Solve 10-4 =1/104 =1/10000
Page 1 of 1
N5/N3=N2
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.
^n=To the power of n - R=root
Cameron Parker
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

^n=To the power of n - R=root
Cameron Parker
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

^n=To the power of n - R=root
Cameron Parker
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

^n=To the power of n - R=root
Cameron Parker
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, poweron-power rule and the factor rule, these are:
The multiplication rule:
The division rule:
The power-on-power rule:
The factor rule:
am x an = am+n
am / an = am-n
(am)n = amxn
(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