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.

2³ = 2 x 2 x 2

2⁴ = 2 x 2 x 2 x 2 and of course 2¹ = 2

In general α¹ = α

α² = α x α

α³ = α x α x α

α^п = α x α x α x α …… α, n times.

Laws of Indices

Law 1: α^m x α^п

= α^п+1

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

5³ x 5⁴

= 5³⁺⁴

= 5⁷//.

Laws 2: α^m ÷ αⁿ

= α^m-n

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

7⁵ ÷ 7³

= 7^{5 - 3}

= 7²//

Law 3: (α^m)ⁿ

= α^{m x n}

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

(5 ³)²

= 5 ³ ^x ²

= 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

= 3^{4 - 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

= 4^3-5

= 4^-2

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

4³

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: α^1/n = ⁿ α (for all values of an excluding zero) e.g

i. 9^½ ii. 4^{– ½}

= √9 = 1

4^½

= 3 = 1 1

√4 = 2 or 2^-1//.

Law 7: α^m/n = (n√α)^m or ⁿ√α^m e.g

i. 27²/³ ii. 8²/³

= (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 4^th root, 5^th 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. α^m x αⁿ = α^m+n
2. α^m ÷ αⁿ = α^m-n
3. (α^m)ⁿ = α^{m x n}
4. α^o = 1
5. α^–n = 1

αⁿ

6. α^1/n = √α
7. α^m/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)

= χ^5-2 x y^2-1 (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)

α^4-2 x b^3-1 + α^5-2 x b^6-1 (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

(χ^-2)³

= χ^{-2 x 3}

= χ⁶

5. Simplify 4^-½ x 16^¾

4^½

Solution 4^–½ x 16^¾

4^½

= 1

4^½ x 16^¾ (Applying law 5)

4^½

= 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.