**TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO**

**Exponents**

Exponents tell how many times to use a number itself in multiplication. There are different laws that guides in calculations involving exponents. In this chapter we are going to see how these laws are used.

Indication of power, base and exponent is done as follows:

Solution:

To write the expanded form of the following powers:

Solution

To write each of the following in power form:

Soln.

The Laws of Exponents

List the laws of exponents

**First law:**Multiplication of positive integral exponent

**Second law:Â**Division of positive integral exponent

**Third law:**Â Zero exponents

**Fourth law:Â**Negative integral exponents

Verification of the Laws of Exponents

Verify the laws of exponents

**First law:**Â Multiplication of positive integral exponent

Generally,

when we multiply powers having the same base, we add their exponents.

If x is any base and m and n are the exponents, therefore:

when we multiply powers having the same base, we add their exponents.

If x is any base and m and n are the exponents, therefore:

Example 1

Solution

If you are to write the expression using the single exponent, for example,(6

^{3})^{4}.The expression can be written in expanded form as:Generally if a and b are real numbers and n is any integer,

Example 2

Example 3

Example 4

Generally, (x

^{m})^{nÂ }= X^{(mxn)}Example 5

Rewrite the following expressions under a single exponent for those with identical exponents:

**Second law:Â**Division of positive integral exponent

Example 6

Example 7

Therefore,

to divide powers of the same base we subtract their exponents (subtract

the exponent of the divisor from the exponent of the dividend). That

is,

to divide powers of the same base we subtract their exponents (subtract

the exponent of the divisor from the exponent of the dividend). That

is,

whereÂ

*x*Â is a real number andÂ*x*Â â‰ 0, m and n are integers. m is the exponent of the dividend and n is the exponent of the divisor.Example 8

solution

**Third law:Â**Zero exponents

Example 9

This is the same as:

If a â‰ 0, then

Which is the same as:

Therefore ifÂ

*x*Â is any real number not equal to zero, then X^{0}Â = 1,Note that 0^{0}is undefined (not defined).**Fourth law:**Â Negative integral exponents

Also;

Example 10

Exercise 1

**1.**Â Indicate base and exponent in each of the following expressions:

**2.Â**Write each of the following expressions in expanded form:

**3.Â**Write in power form each of the following numbers by choosing the smallest base:

- 169
- 81
- 10,000
- 625

a. 169 b. 81c. 10 000 d. 625

**4.**Â Write each of the following expressions using a single exponent:

**5.**Â Simplify the following expressions:

**6.**Â Solve the following equations:

**7.**Â Express 64 as a power with:

- Base 4
- Base 8
- Base 2

Base 4 Base 8 Base 2

**8.**Â Simplify the following expressions and give your answers in either zero or negative integral exponents.

**9.Â**Give the product in each of the following:

**10.**Â Write the reciprocal of the following numbers:

Laws of Exponents in Computations

Apply laws of exponents in computations

Example 11

Solution

**Radicals**

Radicals

are opposite of exponents. For example when we raise 2 by 2 we get 4

but taking square root of 4 we get 2. The same way we can raise the

number using any number is the same way we can have the root of that

number. For example, square root, Cube root, fourth root, fifth roots

and so on. We can simplify radicals if the number has factor with root,

but if the number has factors with no root then it is in its simplest

form. In this chapter we are going to learn how to find the roots of the

numbers and how to simplify radicals.

are opposite of exponents. For example when we raise 2 by 2 we get 4

but taking square root of 4 we get 2. The same way we can raise the

number using any number is the same way we can have the root of that

number. For example, square root, Cube root, fourth root, fifth roots

and so on. We can simplify radicals if the number has factor with root,

but if the number has factors with no root then it is in its simplest

form. In this chapter we are going to learn how to find the roots of the

numbers and how to simplify radicals.

When

a number is expressed as a product of equal factors, each of the

factors is called the root of that number. For example,25 = 5Ã— 5;so, 5

is a square root of 25: 64 = 8Ã— 8; 8 is a square root of 64: 216 = 6 Ã—6

Ã—6, 6 is a cube root of 216: 81 = 3 Ã— 3 Ã—3 Ã—3,3 is a fourth root of 81:

1024 = 4 Ã—4 Ã—4 Ã—4 Ã—4, 4 is a fifth root of 1024.

a number is expressed as a product of equal factors, each of the

factors is called the root of that number. For example,25 = 5Ã— 5;so, 5

is a square root of 25: 64 = 8Ã— 8; 8 is a square root of 64: 216 = 6 Ã—6

Ã—6, 6 is a cube root of 216: 81 = 3 Ã— 3 Ã—3 Ã—3,3 is a fourth root of 81:

1024 = 4 Ã—4 Ã—4 Ã—4 Ã—4, 4 is a fifth root of 1024.

Therefore, the nth root of a number is one of the n equal factors of that number. The symbol for nth root is

whereâˆšis called a radical and n is the index (indicates the root you

have to find). If the index is 2, the symbol represents square root of a

number and it is simply written asâˆšwithout the index 2.

^{n}âˆšwhereâˆšis called a radical and n is the index (indicates the root you

have to find). If the index is 2, the symbol represents square root of a

number and it is simply written asâˆšwithout the index 2.

^{n}âˆšpis expressed in power form as,**nth root of a number by prime factorization**

Example 1, simplify the following radicals

Soution

Basic Operations on Radicals

Perform basic operations on radicals

Different operations like addition, multiplication and division can be done on alike radicals as is done with algebraic terms.

Simplify the following:

Solution

Simplify each of the following expressions:

group the factors into groups of two equal factors and from each group take one of the factors.

The Denominator

Rationalize the denominator

If

you are given a fraction expression with radical value in the

denominator and then you express the expression given in such a way that

there are no radical values in the denominator, the process is called

rationalization of the denominator.

you are given a fraction expression with radical value in the

denominator and then you express the expression given in such a way that

there are no radical values in the denominator, the process is called

rationalization of the denominator.

Example 12

Rationalize the denominator of the following expressions:

Example 13

Rationalize the denominator for each of the following expressions:

Solution

To

rationalize these fractions, we have to multiply by the fraction that

is equals to 1. The factor should be considered by referring the

difference of two squares.

rationalize these fractions, we have to multiply by the fraction that

is equals to 1. The factor should be considered by referring the

difference of two squares.

Exercise 2

**1**. Simplify each of the following by making the number inside the radical sign as small as possible:

Square Roots and Cube Roots of Numbers from Mathematical Tables

Read square roots and cube roots of numbers from mathematical tables

If

you are to find a square root of a number by using Mathematical table,

first estimate the square root by grouping method. We group a given

number into groups of two numbers from right. For example; to find a

square root of 196 from the table, first we have to group the digits in

twos from right i.e. 1 96. Then estimate the square root of the number

in a group on the extreme left. In our example it is 1. The square root

of 1 is 1. Because we have two groups, this means that the answer has

two digits before the decimal point. Our number is 196, read 1.9 in the

table on the extreme left. From our number, we are remaining with 6, now

look at the column labeled 6. Read the number where the row of 1.9 meet

the column labeled 6. It meets at 1.400. Therefore the square root of

196 = 14.00 since we said that the answer must have2 digits before

decimal points. Note: If you are given a number with digits more than 4

digits. First, round off the number to four significant figures and then

group the digits in twos from right. For example; the number 75678 has

five digits. When we round it off into 4 digits we get 75680 and then

grouping into two digits we get 7 56 80. This shows that our answer has 3

digits. We start by estimating the square root of the number to the

extreme left, which is 7, the square root of 7 is between 2 and 3. Using

the table, along the row 7.5, look at the column labeled 6. Read the

answer where the row 7.5 meets the column 6. Then go to where it is

written mean difference and look at the column labeled 8, read the

answer where it meets the row 7.5. Take the first answer you got where

the row 7.5 met the column 6 and add with the second answer you got

where the row 7.5 met the column 8 (mean difference column). The answer

you get is the square root of 75680. Which is 275.1

you are to find a square root of a number by using Mathematical table,

first estimate the square root by grouping method. We group a given

number into groups of two numbers from right. For example; to find a

square root of 196 from the table, first we have to group the digits in

twos from right i.e. 1 96. Then estimate the square root of the number

in a group on the extreme left. In our example it is 1. The square root

of 1 is 1. Because we have two groups, this means that the answer has

two digits before the decimal point. Our number is 196, read 1.9 in the

table on the extreme left. From our number, we are remaining with 6, now

look at the column labeled 6. Read the number where the row of 1.9 meet

the column labeled 6. It meets at 1.400. Therefore the square root of

196 = 14.00 since we said that the answer must have2 digits before

decimal points. Note: If you are given a number with digits more than 4

digits. First, round off the number to four significant figures and then

group the digits in twos from right. For example; the number 75678 has

five digits. When we round it off into 4 digits we get 75680 and then

grouping into two digits we get 7 56 80. This shows that our answer has 3

digits. We start by estimating the square root of the number to the

extreme left, which is 7, the square root of 7 is between 2 and 3. Using

the table, along the row 7.5, look at the column labeled 6. Read the

answer where the row 7.5 meets the column 6. Then go to where it is

written mean difference and look at the column labeled 8, read the

answer where it meets the row 7.5. Take the first answer you got where

the row 7.5 met the column 6 and add with the second answer you got

where the row 7.5 met the column 8 (mean difference column). The answer

you get is the square root of 75680. Which is 275.1

**Transposition of Formula**

Re-arranging Letters so that One Letter is the Subject of the Formula

Re-arrange letters so that one letter is the subject of the formula

A formula is a rule which is used to calculate one quantity when other quantities are given. Examples of formulas are:

Example 14

From the following formulas, make the indicated symbol a subject of the formula:

Solution

Transposing a Formulae with Square Roots and Square

Transpose a formulae with square roots and square