• UNIT1: INDICES AND SURDS





    Key unit competence

    By the end of this unit, I will be able to:

    • Calculate with indices and surds.

    • Use place value to represent very small and very large numbers.

    Unit outline

    • Definition of indices

    • Properties of indices

    • Simple equation involving indices

    • Standard form

    • Definition and examples of surds/ radicals

    • Properties, simplification and operation of surds

    • Rationalization of denominator

    • Square root calculation methods

    Introduction

    Most of our daily activities involve writing very large numbers or very small numbers. For example 1 500 000 and 0.00 001 251. Writing these numbers repeatedly is tedious and in most cases can lead to errors of omission of zeros or other digits. To avoid this, the numbers are therefore written in index form or in standard form. In this unit, we will be writing numbers in index notation and in standard form.

    1.1Indices

    1.1.1 Index notation

    Activity 1.1

    1. Write the following numbers as products of their prime numbers. (a) 16 (b) 81

    2. Discuss with your classmate then express the factors of the numbers in simple form.

    3. Compare your results with another classmates.

    Consider the number 32. Writing it as a product of its prime numbers we get 32 = 2 × 2 × 2 × 2 × 2

    We notice that in this format, the factor 2 is repeated 5 times. We can write the same in short form as:

                                       

    The raised numeral is called an index (plural indices), power or exponent. Representing a number in this short form is known as index notation.

    For example, 16 can be factorised using a factor tree as shown below.

    Example 1.1

    Write each of the following in its simplest index form.

    (a) 81     (b)  96      (c)  5 × c × c × 5 × c × 5

    Exercise 1.1

    1. Write each of the following in index form using the specified base. (a) 25 (base 5)

    (b) 64 (base 4)

    (c) 49 (base 7)

    (d) 1 000 (base 10)

    2. Write each of the following in its simplest index form.

    (a) 2 × a × a × a 

    (b) 3 × y × y

    (c) h × h × h × 7 × h × 21

    (d) 3 × b × b × a × b × b × b

    (e) 3 × a × 3 × a × a × a

    3.Write each of the following in its simplest index form.

    (a) 2                   (b) 8 

    (c) 32                  (d) 16 

    (e) 64                    (f) 128

    7. Express 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5 × 5 in index form.

    8. Express 72 and 108 as products of powers of 2 and 3.

    1.1.2 Properties of indices

    1.1.2.1 Multiplication law of indices

    Activity 1.2

    1. Write the following numbers as products of two numbers where the two numbers are not equal or where one of the numbers is not one.  E.g 16 = 2 × 8          

    (a) 32    (b) 81

    2. Write the short form of the prime products of the numbers you wrote.

    3. Discuss with your partner the relationship between the index of the products and the indices of the numbers. 4. Compare your answer with other classmates.

                          

    Generally,

    When numbers, written in index form with a common base, are multiplied, the indices are added while the base remains the same,

    If in a multiplication there is more than one letter to be multiplied, they must be multiplied separately because each represents a different value.

    Squaring an expression simply means multiplying the expression by itself.  For example,

    Thus, in order to square an algebraic expression, square  the base and double the indices of the letters.

    1.1.2.2 Division laws of indices

    Activity 1.3

    1.  Write the following numbers as quotients of two numbers where the two numbers are not equal, the dividend is greater than the divisor and none of them is equal to 1. 

    (a) 4  (b) 3

    2. Write the short form of the quotient, divisor and the dividend in index notation.

    3. Discuss with your partner the relationship between the index of the quotient and that of the indices of the dividend and divisor.

    4. Compare the answers with your partner.

    In the above working, the power of 2 in the answer is the difference between indices of the numerator and that of the denominator.

    Example 1.5


    If there is more than one variable to be divided, they must be divided separately as they represent different values.

    1.1.2.3 Power of powers 

    Activity 1.4

    1. Write the following numbers in index notation.

    (a) 4                     (b) 27

    2. Square each of the numbers.

    3. Find the relationship between the indices of the squares and the solution.

    When a number, written in index form, is raised to another power, the indices are multiplied.

    All the numerals which are multiplied together in a bracket are raised to the power of that bracket

    Note that in this example, all the numerals which are in the bracket are raised to the power of the bracket

    1.1.2.4 Zero index

    Activity 1.5

    1. Using the division law of indices, find the solution of the following:

    2. Discuss the results with your partner and compare them with other classmates.

    1.1.2.5 Negative indices

    Activity 1.6

    1. Using the division law of  indices, solve the following:

    2. In the second case, express your answer in index notation and in fraction form.
    3. Discuss with your classmates the relationship between the answer in index notation and fraction form.

    Any number raised to a negative power is the same as the reciprocal of the equivalent positive power of the same number,

    1.1.2.6 Fractional indices
    Ac

    1.1.2.6 Fractional indices

    Activity 1.7

    2. Discuss the changes in the index and how in turn it affects the results.

    3. Derive a relationship between the change in index and the result. Thus predict the result of 31 2.

    4. Compare your results with other classmates.

    The indices are reducing by
    1 /2 while the results are the square roots of the preceding results: ie

    We can also understand the meaning of fractional indices by applying the laws of indices. For example, we know that fractional powers obey the same laws of indices as integral powers.  What then is the meaning of:


    Note that method (b) is quicker.

    From Example 1.10, we see that to find the square root of an algebraic expression, we find the square root of the coefficient and divide the indices of the letters by 2.

    Similarly, for any root, order n, we simply find the nth root of the coefficient and divide the indices of the letters by the order n of the root.


    1.1.3 Simple equations  involving    indices

    Activity 1.8

    1. Consider the equation  = 4

    2. Discuss in groups how you can determine the value of x.

    3. Compare your findings with another groups

    Sometimes we may be required to solve equations involving indices.

    Consider = 81.  What value of x makes the equation true? 

    =  81 is the same as =  34 (expressing Right Hand Side (RHS) is index form).

    The base on the Left Hand Side (LHS) is equal to the base on the RHS.  Since LHS = RHS, the indices must also be equal.

    ∴ if   =  34,  then       x   =  4  (equating indices since bases
            are  also equal).

    In general,

    If then x = y Similarly,  if, then a = b, provided both the bases are positive numbers. To solve equations involving indices, follow the procedure below:

    1. Express both sides of the equation with a common base and simplify as far as possible to reduce to one term on LHS and one term on RHS.

    2. If the variable is in the exponent, equate the indices and solve the resulting equation.

    3. If the variable is in the base, ensure that the powers are the same. Equate the bases and solve the resulting equation.

    1.2  Standard form

    Activity 1.9

    1. Write the following numbers as products of two numbers. Where one of the numbers is either between 1 and 10 (10 not included) while the other number should be a power of 10)

    (a) 1 000        (b) 100 000

    (c) 10              (d) 1

    (e) 0.001             (f)  7 000

    2. Compare your results with other classmates.

    Consider the number 60 000. We can write this number using the instructions in step 1 of activity 1.9 as follows:

    60 000 = 6 × 10 000

    Where 10 000 in index notation is 

    10 × 10 × 10 × 10 =

    Therefore,

    60 000 = 6 ×

    This format of writing is known as standard form or scientific notation.

    It involves writing large numbers in terms of powers of 10.

    Example 1.17

    Write each the following numbers in scientific notation.

    (a) 75 200                (b) 0.000321

    (c) 85                       (d) 7 834 

    (e) 7.321                 (f) 0.0429

    Solution

    (a)  75 200: In this case, our decimal point is at the end of the number to the right. To write this number in scientific notation, we move the decimal place to the left up to the position of the number in the 1st significant place value. We then count the number of steps we have moved the decimal point and write it as a power of 10.
     

    (b) 0.000321: In case of a decimal number, we use the same procedure as in (a) above but this time, our decimal point moves to the right. 10 will now be raised to a negative power (ie number of steps moved to the right).
     

    Exercise 1.9 1.

    Write the following numbers in standard notation.

    (a) 601                 (b) 42 300

    (c) 6 001               (d) 4 329 200

    (e) 100 000 000      (f) 75 000

    (g) 0.000 561           heart 0.000 000 32

    2. Multiply each of the following and leave your answer in standard form.

    (a) 326 × 43           (b) 41 × 691

    (c) 8.5 × 25              (d) 69 × 7

    (e) 6 300 × 90            (f) 55 × 20

    (g) 439 × 12                heart 640 × 15

    1.3 Surds

    1.3.1 Definition of a surd

    Activity 1.10

    1. On a stiff paper such as manilla paper, construct two squares of sides 1 unit each. (You may use a scale of "4 cm represents 1 unit"). 2. Cut each square diagonally and rearrange the triangles to form one large square as shown in Fig. 1.1.


    3. The sides of the larger square are equal in length to the diagonals of the smaller squares. What is the length?

    4. What is the exact area of the large square? Now calculate the area using the length of the side that you measured in step 3 above. Did you get 1.96?

    From Activity 1.10, we see that: Area of large square = sum of areas of 2 small squares

    Many numbers are not exact powers.  Their roots (e.g. square root, cube root, etc.) are, therefore, irrational.  Expressions containing roots of such numbers are called surds.

    1.3.3  Simplification of surds

    Activity 1.11

    By putting x = 25 and y = 4, determine which of the following pairs of expression are equal.

    Note that in Activity 1.11, the pairs of expressions in 1, 2 and 6, are equal.

    These facts can be used when simplifying surds.

    A surd is said to be in its simplest form when the number under the radical is a prime number.

    The process of simplifying surds can also be reversed to obtain surds of single numbers.

    1.3.4 Operation on surds

    1.3.4.1 Addition and subtraction of surds

    Activity 1.12

    1. Using the basic addition and subtraction mathematics operations, calculate the following:

    (a) x -x         (b)3x-2x

    We notice that:

    To be able to add or subtract surds, they must contain roots of the same number. In general,

    1.3.4.2 Multiplication of surds

    Activity 1.13

    When two monomial surds have to be multiplied together;

    1.  first simplify each surd where possible, and then

    2.  multiply whole numbers together and surds together.

    Note that the second method is sometimes simpler and quicker.

    1.3.4.3 Division of surds and rationalising the denominator

    Activity 1.14

    If a fraction has a surd in the denominator, it is usually better to rationalise the denominator.

    Rationalising the denominator means making the denominator a rational number, so that we divide by a rational number rather than divide by a surd.  When rationalising, we multiply both the numerator and the denominator of the fraction by a surd which makes the denominator rational.  It is easier to divide by a rational number than a surd.

    Rationalisation of monomial denominators

    Activity 1.15

    1. Using the knowledge of multplication of surds, discuss with your patner how to rationalise the denominator of:

    2. Compare your findings with other classmates.

    Rationalisation of binomial denominators

    Activity 1.16

    Note that 1 is a rational number. Any two surds whose product give a rational number are called  conjugate surds.

    Therefore, to rationalise a denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.

    NB

    When adding or subtracting fractions containing surds, it is advisable to first rationalise the denominator of each fraction. This is done by multiplying the numerator and the denominator of the fractions by the same number so that the answer has a rational denominator.

    If the product of two surds is a rational  number, then the surds are said to be conjugates of each other or simply conjugate surds

    The first method gives an “exact” answer while the second one gives an approximation.  Thus, the first method is more accurate

    Note:  You may use a calculator when evaluating surds, but you still must show all the steps involved in the process.

    1.4 Square roots 1.4.1

    Square root by estimation method

    Activity 1.17


    Consider the value of   . Since is not a perfect square, 10    is also not an exact value. 10 lies between 9 and 16 both of which are perfect squares i.e

    This tells us that   is a value between 3 and 4. We can obtain the value by estimation method. The following example will help us understand how to do this.

    NB

    To find the whole number part of a square root of a non–square number,

    1.  Find two consecutive exact squares between which the non-square number lies, (e.g. 39 lies between 36 and 49).

    2.  The square root of that number lies between the square roots of the exact squares.

    3.  The whole number part of the required square root is given by the lower value,

    Exercise 1.13

    1.  The following numbers are exact squares.

    Find their positive square roots.

    (a)  64      (b) 144  

    (c) 400     (d) 169 

    (e) 1.96     (f) 0.0036

    2.  Find the two consecutive exact squares closest integers between which the square root of each of the following numbers lies.

    (a) 134       (b)  430     

    (c) 1 440     (d) 3 000

    1.4.2. Square root by factorisation

    Activity 1.18

    To find the square root by factorisation

    (a)   Express the given number as a product of prime factors in power notation.

    (b)  Divide the power of each factor by 2.

    (c)  Multiply out the result to get the square root.

    1.4.3 Square root by general method

    First, the numbers under the root are grouped in pairs from right to left. Make sure you leave at least one or two digits on the left. For each pair of numbers you will get one digit in the square root. To start, find a number whose square is less than or equal to the first pair or first number, and write it above the square root line. 

                         

    2. Surds: These are expressions containing roots of irrational numbers.

    3. Scientific notation: It involves writing large numbers in terms of powers of 10 in the form A x where the index n is a positive or negative integer and A must lie in the range

    1 ≤ A ≤10.

    4. Monomial surd: It is a surd which contains only one term.

    5. Binomial surd: It is a surd which has two terms.

    6. Rationalising the denominator: It means making the denominator a rational number by multiplying both the numerator and denominator by the conjugate of the denominator.

    7.  Conjugate surds: Occurs when the product of two surds is a rational number.

             Unit 1 Test

UNIT2: POLYNOMIALS