FORM TWO MATHEMATICS STUDY NOTES TOPIC 1-2.

TOPIC 1: EXPONENTS AND RADICALS

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: Division of positive integral exponent

Third law: Zero exponents

Fourth law: Negative integral exponents

Verification of the Laws of Exponents

Verify the laws of exponents

First law: 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:

Example 1

Solution

If you are to write the expression using the single exponent, for example,(6³)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)ⁿ= X^(mxn)

Example 5

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

Second law: 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,

where x is a real number and x ≠ 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: Zero exponents

Example 9

This is the same as:

If a ≠ 0, then

Which is the same as:

Therefore if x is any real number not equal to zero, then X⁰ = 1,Note that 0⁰is undefined (not defined).

Fourth law: Negative integral exponents

Also;

Example 10

Exercise 1

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

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

3. 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. Write each of the following expressions using a single exponent:

5. Simplify the following expressions:

6. Solve the following equations:

7. Express 64 as a power with:

Base 4
Base 8
Base 2

Base 4 Base 8 Base 2

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

9. Give the product in each of the following:

10. 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.

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 = 5× 5;so, 5 is a square root of 25: 64 = 8× 8; 8 is a square root of 64: 216 = 6 ×6 ×6, 6 is a cube root of 216: 81 = 3 × 3 ×3 ×3,3 is a fourth root of 81: 1024 = 4 ×4 ×4 ×4 ×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 isⁿ√ where√is 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 as√without the index 2.ⁿ√pis 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.

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.

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

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

Make the indicated symbol a subject of the formula:

Exercise 3

1. Change the following formulas by making the given letter as the subject of the formula.

2. Use mathematical tables to find square root of each of the following:

When we play games with computers we play by running, jumping and or finding secret things. Well, with Algebra we play with letters, numbers and symbols. And we also get to find secret things. Once we learn some of the ‘tricks’ it becomes a fun challenge to work with our skills in solving each puzzle. So, Algebra is all about solving puzzles. In this chapter we are going to learn some of the skills that help in solving mathematics puzzles.

Binary Operations

The Binary Operations

Describe the binary operations

When two numbers are combined according to the instructions given and produce one number we say that they are binary operations. For example, when we add 4 to 6 we get 10, or when we multiply 4 by 3 we get 12. We see that addition of two numbers lead to one number and multiplication of two numbers produce one number. This is binary operation. The instructions can be given either by symbols like X,*,∇and so on or by words.

Performing Binary Operations

Perform binary operations

Example 1

Evaluate:

solution

Example 2

Find

solution

Example 3

Solve

Example 4

evaluate,

Example 5

Calculate

Brackets in Computation

Brackets are used to group items into brackets and these items inside the brackets are considered as whole. For example,15 ÷(X + 2) ,means that x and 2 are added first and their sum should divide 15. If we are given expression with mixed operations, the following order is used to perform the operations: Brackets (B) are opened (O) first followed by Division (D) then Multiplication (M), Addition (A) and lastly Subtraction (S). Shortly is written as BODMAS.

Basic Operations Involving Brackets

Perform basic operations involving brackets

Example 6

Simplify the following expressions:

4 + 2b – (9b ÷3b)
4z – (2x + z)

solution

Algebraic Expressions Involving the Basic Operations and Brackets

Simplify algebraic expressions involving the basic operations and brackets

Example 7

Evaluate the following expressions:

solution

Identities

For example, 3(2y + 3) = 6y + 9, when y = 1, the right hand side (RHS) and the left hand side (LHS) are both equals to 15. If we substitute any other, we obtain the same value on both sides. Therefore the equations which are true for all values of the variables on both sides are called Identities. We can determine whether an equation is an identity or not by showing that an expression on one side is identical to the other expression on the other side.

Example 8

Determine whether or not the following expressions are identities:

solution

Quadratic Expressions

A Quadratic Expression from Two Linear Factors

Form a quadratic expression from two linear factors

A quadratic expression is an expression where the highest exponent of the variable (usually x) is a square (x²). It is usually written as ax²+bx+c.

Activity 1

Form a quadratic expression from two linear factors

The General Form of Quadratic Expression

Write the general form of quadratic expression

Quadratic expression has the general form of ax² + bx + c where a ≠ 0 and a is a coefficient of x² , b is a coefficient of x and c is a constant. its highest power of variable is 2. Examples of quadratic expressions are 2x² + x + 1, 4y² + 3, 3z² – 4z + 1 and so on. In a quadratic expression 3z² - 4z + 1, a = 3, b = -4 and c = 1. Also in quadratic expression 4y² + 3, a = 4, b = 0 and c = 3

Example 9

If you are told to find the area of a rectangle with a length of 4y + 3 and a width of 2y + 1.

Solution

Example 10

3x items were bought and each item costs (4x – 3) shillings. Find total amount of money used.

Factorization

Linear Expressions

Factorize linear expressions

The operation of resolving a quantity into factors, when we expand expressions, is done by removing the brackets. The reverse operation is Factorizing and it is done by adding brackets.

Example 11

Factorize the expression 5a+5b.

Solution

In factorization of 5a+5b, we have to find out a common thing in both terms. We can see that the expression 5a+5b, have got common coefficient in both terms, that is 5. So factoring it out we get 5(a+b).

Example 12

Factorize 18xyz-24xwz

Solution

Factorizing 18xyz-24xwz, we have to find out highest common factor of both terms. Then factor it out, the answer will be 9xz(2y-3w).

Quadratic Expressions

Factorize quadratic expressions

When we write the quadratic expression as a product of two factors we say that we have factorized the expression. We are going to learn two methods used to factorize quadratic expressions. These methods are factorization by Splitting the middle term and factorization by Inspection.

Factorization by splitting the middle term