MTH5P1:Surds, Indices, and Logarithms

Radical
Definition of the Radical
For all real x, y > 0 , and all integers a > 0 ,

A number which can be expressed as a fraction of integers (assuming the denominator is never 0)

is called a rational number. Examples of rational numbers are 5/2, -4/5 and 2.

A number which cannot be expressed as a fraction of two integers is called an irrational number.

Examples of irrational numbers are 

A number which can be expressed as a fraction of integers (assuming the denominator is never 0) is called a rational number. Examples of rational numbers are 5/2, -4/5 and 2.

An irrational number involving a root is called a surd. Surds occur frequently in trigonometry,
calculus and coordinate geometry. Usually, the exact value of a surd cannot be determined but an
approximate value of it can be found by using calculators or mathematical tables. In this chapter,

General Rules of Surds
Multiplication of surds

Division of surds

These rules are useful for simplifying two or more surds of for combining them into one single
surd.

Rationalization of the Denominator.

When a fraction has a surd in its denominator, e.g. 

it is usual to eliminate the surd in the denominator. In fact, the writing of surds in the denominators of fractions should be avoided.
The process of removing this surd is called rationalizing of the denominator.

are specially related surds known as conjugate surds. The product of conjugate surds is always a rational number.

For example

Example 3

Example 4

Solution:

Simplify

Example 5

Simplify, without using tables or calculators, the value of

Indices

If a positive integer a is multiplied by itself three times, we get

means the nth power of a, where n is any positive index of the positive integer a.

Rules of Indices
There are several important rules to remember when dealing with indices.
If a, b, m and n are positive integers, then

Solving Exponential Equations
Example 3
Solve the following exponential equations

Example 5

Logarithms
Definition:

Laws of Logarithms

Changing the Base of Logarithms

Laws of Logarithms

Changing the Base of Logarithms