Principle of Indices Index exponential power Base Indices

Principle of Indices Index, exponential, power Base

Indices Rule 1

Indices Rule 2

Indices Rule 3 From Rule 1

Indices Rule 4

Indices Rule 5

Indices Rule 6, 7 1

Indices Rule 8

Irrational Number that cannot be expressed as a fraction of two integers

Surd Rules We can use the above rules to: • simplify two or more surds or • combining them into one single surd

Example 1: Simplify the following surds : Solution:

Rationalization of the Denominator Process of removing a surd from the denominator Example 2: Solution: Multiple together!

Simplifying Indices Change to common base Change to common power Break each term into its prime factor Think of how to make it common power Simplify the indices within each term Ensure that common power Simplify the indices across other terms Combine the base and simplify

Quadratic Equation - Definitions (Expression & Equation) Expression: Representation of relationship between two (or more) variables Y= ax 2+bx+c, Equation : Statement of equality two expression ax 2 + bx + c between = 0 Root: -value(s) for which a equation satisfies Example: x 2 -4 x+3 = 0 (x-3)(x-1) = 0 x = 3 or 1 Roots satisfies of x 2 -4 x+3 = 0

Quadratic Equation Definitions (Quadratic & Roots) Quadratic: A polynomial of degree=2 y= ax 2+bx+c = 0 is a quadratic equation. (a 0 ) A quadratic equation always has two roots

Quadratic Equation -Factorization Method Solve for x 2+x-12=0 Step 2: factors Step 1: product -12 -4, 3 -2, 6 4, -3 Step 3: x 2+(4 -3)x -12=0 Roots are -4, 3 factors with opposite sign Sum of factors -1 4 1 x 2+4 x-3 x-12=0 (x+4)(x-3)=0

Quadratic Equation -Factorization Method x 2+x-12=0 x 2+(4 -3)x -12=0 (where roots are – 4, 3) Similarly if ax 2+bx+c=0 has roots , ax 2+bx+c a(x 2 -( + )x + ) Comparing co-efficient of like terms:

Properties of Roots Quadratic equation ax 2+bx+c=0 , a, b, c R and The equation becomes: a { x 2+ (b/a)x + (c/a) }= 0 a x 2 -( + )x+ =0 a(x- )=0 x 2 -(sum) x+(product) =0

General Solution (b 2 - 4 ac) discriminant of the quadratic equation, and is denoted by D. Roots are This is called the general solution of a quadratic equation

Nature of Roots Discriminant, D=b 2 -4 ac D > 0 is real Roots are real a, b, c are rational (D is perfect square) Rational D = 0 D < 0 (D is not a perfect square) Irrational Roots are real and equal is not real Roots are imaginary

Logarithms Definition Base: Any postive real number other than one Log of N to the base a is x Note: log of negatives and zero are not Defined in Reals

Fundamental laws of logarithms

Other laws of logarithms Change of base Where ‘a’ is any other base

System of logarithms Common logarithm: Base = 10 Log 10 x, also known as Brigg’s system Note: if base is not given base is taken as 10 Natural logarithm: Base = e Logex, also denoted as lnx Where e is an irrational number given by