Factor Rule: (a x b)m = am x bm Multiplication Rule: am x an = am+n a-n = 1/an a0 = 1

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1 Nathan Thomas Revision Notes: Index Notation: Rules: Factor Rule: (a x b) m = a m x b m Multiplication Rule: a m x a n = a m+n a -n = 1/a n a 0 = 1 Index Notation Power-on-power Rule: (a m ) n = a mxn Division Rules: a m a n = a m-n Fractional Indices: a 1/2 = a a 2/3 = ( 3 a) 2 = 3 (a) 2 Power a p/q = Root This is (Provided m>n). Zero and negative indices: If m =0 then the multiplication rule no longer applies. E.g. a 0 = 1. If m = -2 (any negative number) then the multiplication rule doesn t apply again. E.g. a -n = 1/a n. This is known as the negative power rule.

2 Index Notation and graphs Apply the rules! The multiplication rule: The division rule: The power on power rule: The factor rule: ( ) ( ) Graph of y= x -1 Graph of y=x 2/3 Graph of y=x 1/2 Graph of y= x 3

3 Thorsten Bell Index Notation -n -m =1/(-n) m x 1/2 =rootx x 0 =1 0 0 =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 x n =A is x=a 1/2 (-a) m =+a m if m is an even integer or zero -a m if m is a negative integer If f(x)=1/x m, 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

4 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

5 Ben Smith 12J Index notation 3x -2 = 3 x 2 (n 2 ) 3 =n 6 Index Notation N 0 = 1 3c 2 * 5c 4 = 15c 6 (ab) m = a m *a m X 1/2 = square root of x Solve 2-3 =1/2 3 =1/8 Solve 10-4 =1/10 4 =1/10000 N 3 * N 4 =n 7 N 5 /N 3 =N 2 Page 1 of 1

6 Indices Cameron Parker ^n=to the power of n - R=root

7 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

8 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

9 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 = ^n=to the power of n - R=root Cameron Parker

10 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

11 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 x 2 being used all the time as shorthand for quadratic equations. In general a simple index notation can given the formula: a m = 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: a m x a n = a m+n a m / a n = a m-n (a m ) n = a mxn (a x b) m = a m x b m 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 a m. If you compare the answers of 2 2 with 2-1 and 2 2 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/a m When the index is zero the answers also follow a pattern in which the answer is always 1, This is formulated as: a 0 = 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: (a 1/2 ) 2 = a 1/2x2 = a 1 Therefore a 1/2 would have to be a number whose square is a. There are only two numbers which equal this; + a or - a. So we say a 1/2 = a. This works for other simple fractions and creates the formula: a 1/m = m a (note that m is the m th 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: a n/m = (m a) n = m a n

A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers.

LEAVING CERT Honours Maths notes on Algebra. A polynomial expression is the addition or subtraction of many algebraic terms with positive integer powers. The degree is the highest power of x. 3x 2 + 2x