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Mathematics Made Easy

Introduction

Many students have already christened mathematics “difficult”. But is mathematics truly all that difficult? The answer is indisputably NO! The Mark education pages, in an attempt to reduce students’ mathematics phobia, introduce a new column, Mathematics Made Easy, with effect from this week. This project will, among other things, drastically curb the rate of mathematics exams failure among secondary students as a first class Mathematics degree holder, POLYCARP DICKSON take students and teachers alike through the West African Examination Council (WAEC)/ National Examination Council (NECO) syllabus.

His words: “I will also advise students in colleges of education and polytechnics particularly those studying science and engineering courses to adhere strictly to this column every week. The plan here is to cover WAEC/NECO syllabus at the end of one year. So if any secondary school students follow the column seriously, at the end of one year, he or she must have gotten a complete mathematics textbook”.

All enquiries on any of the questions solved or comment/criticisms should be directed to:

  1. Polycarp Dickson, email: polycarpdicks2007@yahoo.com or enroute2happy4poly@yahoo.co.uk
  2. The Mark, P.O. Box 319, along Gujba Road, Damaturu, Yobe State. 08036161640, 08026797440.
  3. c/o Pastor Joel Joseph Thiliza, Centre of Grace Church, New Jerusalem, Damaturu. 0806 030 3442.
  4. Polycarp Dickson, Leaders Private School, Gujba Road, Damaturu, Yobe State.

INDICES

Hi students, are you with me? Alright, let’s start with Indices.

Definition:

Indices is the plural form of index and index means power. Therefore, indices means powers. The index notation is used as shorthand for the further related multiplication of the number by itself e.g.

2 = 2 x 2 x 2

24 = 2 x 2 x 2 x 2 and of course 2 = 2

In general α = α

α = α x α

α = α x α x α

αп = α x α x α x α …… α, n times.

Laws of Indices

Law 1: αm x αп

= αп+1

This law, theoretically, means to multiply powers of a number, add the indices e.g.

5 x 54

= 53+4

= 57//.

Laws 2: αm αn

= αm-n

This law, theoretically, states; for the division of the powers of the same number, the indices are subtracted e.g.

75 73

= 75 - 3

= 72//

Law 3:m)n

= αm x n

Theoretically, indices are multiplied when the power of a number is itself raised to another power e.g.

(5 )

= 5 x

= 56

THE ZERO POWER

Law 4: n = 1

Theoretically, any number raised to the power of Zero is equals to one. E.g.

34 34

= 34 - 4

= 3

= 1

NEGATIVE INDICES

Law 5: α-n

= 1

αn

The result of dividing 43 45 using the laws of indices is;

43 45

= 43-5

= 4-2

If we use the notation of the division line, we can see what negative index means at once.

4

45

= 41 x 41 x 41

14 x 14 x 14 x 4 x 4

= 1

42

= 4-2//.

The negative sign is not an indication that the number is negative, but simply that a certain power of number has been divided by a larger power.

FRACTIONAL INDICES

Law 6: α1/n = n α (for all values of an excluding zero) e.g

i. 9 ii. 4

= √9 = 1

4

= 3 = 1 1

√4 = 2 or 2-1//.

Law 7: αm/n = (n√α)m or n√αm e.g

i. 27/ ii. 8/

= (3√27)2 = (3√8)2

= 32 = 22

= 9//. = 4

Note: In law 6, the square root of a certain number means, the digit that can multiply itself two times to obtain the number in the radical sign. Likewise in law 7, the cube root of a number is the digit that can multiply itself three times to obtain the number in the radical sign. Same thing applies to both 4th root, 5th root etc.

Now that you’ve comprehended the laws of indices, let’s proceed to solve some problems based on these laws. I want the student to understand that he cannot solve any indices problem without comprehending the laws. However, my advice to my students is this; go through the laws over and over until you can memorise them without this paper. The laws are stated below again more clearly for your comprehension.

Laws of Indices

  1. αm x αn = αm+n
  2. αm αn = αm-n
  3. m)n = αm x n
  4. αo = 1
  5. α–n = 1

αn

  1. α1/n = √α
  2. αm/n = (n√α)n or n√αn//.

EXAMPLES

1. Simplify α2 b3 x α4 b5

Solution

α2 b3 x α4 b5

= α2 x α4 x b3 x b5 (collecting like terms)

= α2+4 x b3+5 (applying law 1)

= α6 b8//.

2. Simplify χ5 y2

χ2 y

Solution

χ5 y2

χ2 y

= χ5 χ2 x y2 y1 (collecting like terms)

= χ5-2 x y2-1 (applying law 2)

= χ3 y

or χ5 y2

χ2 y

= χ3 y

3. Simplify α4 b3 + α5 b6

α2 b

Solution

α4 b3 + α5 b6

α2 b

= α4 b3 + α5 b6

α2 b α2 b

α4 α2 x b3 b1 + α5 α2 x b6 (collecting like terms)

α4-2 x b3-1 + α5-2 x b6-1 (applying law 2)

= α2 b2 + α3 b5//.

Or α4 b3 + α5 b6

α2b

= α4 b3 + α5 b6

α2 b α2 b

= α2 b2 + α3 b5//.

4. Simplify (χ-2)3

Solution

-2)3

= χ-2 x 3

= χ6

5. Simplify 4- x 16

4

Solution 4 x 16

4

= 1

4 x 16 (Applying law 5)

4

= 1 x (4√16)3

√4 (Applying law 6 and 7)

√4

= 1/2 x 23 (Removing Radical sign)

2

= 1/2 x 8

2

= 8/2

2

= 4

2

= 2//.

Plateau State University: Matters Arising

News Analysis By Deji Ajayi in Jos

The chairman of the administrative and investigative panel on the establishment of the Plateau state university Bokkos, Professor Dakum Shown, recently submitted the reports of his committee to Gov. Jonah David Jang at the Government House Rayfield Jos.

The content of the reports were far from complementary. Consequently academic activties at the institution have been suspended as the panel recommended a drastic review on the setting up of the institution.

Not surprising, students of the institution on hearing the news wasted little time in expressing their displeasure with the action by embarking on a peaceful protest, barricading the serene Barrakin Ladi-Bokkos road hours later.

Their grouse being that the action according to them is inimical to their academic future. Our correspondent reports that Professor Shown’s indicated that the institution exists on a number of unsustainable frameworks ranging from inadequate staffing to inconsistent funding and questionable student intake.

The panel observed that the issue of funding greatly required a review as modalities put in place by the founders to run the institution were unreliable.

For example, the professor noted that some local governments mandated to contribute a certain quota to the university’s vault were acting contrary to the directive. More so, most of the council authorities have failed to show any sign of compliance.

The panel therefore called for the suspension of academic activities in the school. Prof. Shown pointed to the unreliable teaching staff situation in the school where 55 percent of them were on temporary recruitment and sourced from another institution.

His words: “You would find in our report an interesting thing, the Universtity of Jos was shocked to find out that its staff were being engaged in that university (PLASU), a scenario that leaves the school with no permanent academic staff. More so the temporary staff drawn from Unijos enjoy full pay for part -time services”.

With the institution lacking heads of departments, deans of faculties and other principal academic staff associated with a standard university, the investigative panel defended its recommendations which are currently generating heated argument across the state.

The state government on its part acknowledged receipt of the panel’s report and heeded its counsel to discontinue academic exercise in the school.

In a press statement signed by James Mannok, press secretary to the state governor revealed that “Government wishes to state that it has accepted the recommendations and ordered the suspension of the opening of the university”.

Expectedly, the affected students and their parents are crying foul. They claimed that their pursuit of higher education in the school was as a result of the limited opportunities available in the existing institutions across the state.

One of the parents who spoke to The Mark on the condition of anonymity urged the government to “carefully consider the fate of our children as the higher institutions on ground cannot adequately absorb them”.

Another parent also warned that “the government should not play politics with the education of our wards”.

The institution like many others nationwide was set up to take care of the ever increasing number of school leavers seeking admission into tertiary schools.

YOUNG WRITERS’ WORLD

Why I want to be a doctor

By Mohammed Alhassan Datti

People suffer from various diseases in my country, Nigeria. There are many destitutes and the sick. Yet, there are inadequate healthcare facilities which results to high rate of mortality.

My father happens to be a doctor, an anesthesiologist. Sometime ago, I went along with him to his office at the hospital where I saw a patient who is a young boy of my age. The boy was suffering from a disease known as leukemia. He and his parents were crying profusely because they were unable to raise the bill required by the hospital for his treatment. They spent several hours crying with no help in sight, no one rendered assistance to them; and eventually, the boy died.

His death, especially the event leading to it, touched my heart. It made me sad and I wept too. I felt how bad it is for one to be sick and not be able to raise money for treatment. I know how it hurts for one to lose someone so dear. More so, I know how much I love my parents and how deep their love is for me.

Therefore, I vow that I will grease my elbow, and buckle up at school. I vow that by the grace of God I will become a medical doctor. My desire is not just neither to wear the stethoscope or the white clean coat nor for the title, just to be referred to as DOCTOR. It is equally not for the sake of acquiring wealth; for they are worldly possession, but to help patients who are indigent. To give people the best I can so that Nigeria can be a greater country.

I want to, as a result of the above account of the helpless boy, appeal to Nigerian children to please strive hard to be counted among the good citizens. They should always be positive and see the future as bright. They should not be disenchanted. We should strive to always remember that we are the leaders of tomorrow.

Furthermore, we should know that for whatever good deed we do, there is a reward. Thank God, now you must not wait for the hereafter to get your reward and dues like the teachers in Nigeria used to. Today, the reward is both here and the hereafter when we reach our final destination.

What is more? If wishes were horses, I would have wished I am a doctor now so I could immediately start touching the lives of the people, especially the downtrodden.

 Mohammed Alhassan Datti, is a 12-year-old SSS III student of Assalam Private School Kano, Nigeria

Brief Biodata of Polycarp Dickson

Polycarp Dickson holds a 1st Class Bachelors Degree in Mathematics (B.Sc Hons) Maths from the University of Calabar. He holds a Professional Graduate Diploma (PGO) in Operations Research (OR) from the Pacific State University, New Zealand and a Master of Science (M.Sc.) Degree in Space Technology and Mathematics from Pacific State University New Zealand.

He also bagged the 1st position in Commonwealth Mathematics ABICS competition organised for all the Nigerian Universities in 1998 and 2nd position in the Commonwealth Mathematics competition organised for all the West African countries (2004) in Ghana. He wishes you success as you partake in keeping a date with this column.

                  

Vol. 6 No. 15 November 17, 2007  EDITION...Of Truth and Excellence